Geometry & Topology 10 (2006) 2117–2171
arXiv version: fonts, pagination and layout may vary from GT published version
2117
Zero dimensional Donaldson–Thomas
invariants of threefolds
JUN LI
Using a homotopy approach, we prove in this paper a conjecture of Maulik, Nekrasov,
Okounkov and Pandharipande on the dimension zero Donaldson–Thomas invariants
of all smooth complex threefolds.
14D20; 14J60
0
Introduction
Ever since the pioneer work of Donaldson and Thomas on Yang–Mills theory over
Calabi–Yau threefolds [5, 13], people have been searching for their roles in the study of
Calabi–Yau geometry and their relations with other branches of mathematics. The recent
results and conjectures of Maulik, Nekrasov, Okounkov and Pandharipande [10, 11] that
relate the invariants of the moduli of ideal sheaves of curves on a Calabi–Yau manifold
to its Gromov–Witten invariants constitute major progress in this direction. This paper
will address one of their conjectures on invariants associated to Hilbert scheme of points.
To begin with, we let (X, H) be a smooth projective threefold over complex numbers C.
For any integer r ≥ 0, a line bundle I ∈ Pic(X) and two classes c2 ∈ H 4 (X, Z) and
c3 ∈ H 6 (X, Z), we form the moduli space
MH
X (r, I, c2 , c3 )
of H –stable sheaves of OX –modules E satisfying
rkE = r,
det E = I,
c2 (E) = c2
and
c3 (E) = c3 .
Back in the seventies, Maruyama [9] proved that such moduli spaces are quasi-projective,
and become projective in case the quaduple (r, I, c2 , c3 ) is relatively prime. Later,
Mukai [12] showed that the first order deformations of any sheaf E in these moduli
spaces are given by the traceless part of the extension group Ext1 (E, E); the obstructions
Published: 29 November 2006
DOI: 10.2140/gt.2006.10.2117
2118
Jun Li
to deforming E lie in the traceless part of Ext2 (E, E) (see also Artamkin [1]). In case X
is a Calabi–Yau threefold and E is stable, the traceless part
Ext3 (E, E)0 = Ext0 (E, E)∨
0 =0
while
Ext2 (E, E)0 = Ext1 (E, E)∨
0.
Hence the moduli space MH
X (r, I, c2 , c3 ) admits a perfect-obstruction theory as defined
by Li and Tian [7] and Behrend and Fantechi [4]; when it is projective, it carries a
virtual dimension zero cycle
vir
[MH
∈ H0 MH
X (r, I, c2 , c3 )]
X (r, I, c2 , c3 ), Z .
Its degree is the invariant originally defined and studied by Donaldson and Thomas [5].
Following [10, 11], we shall call them Donaldson–Thomas invariants of the Calabi–Yau
manifold X .
One special class of such moduli space studied extensively in [10, 11] is when r = 1
and I = OX . Because X is smooth and E is stable, it must be torsion free, and be a
subsheaf of its double dual E ∨∨ ∼
= OX ; hence it becomes an ideal sheaf of a subscheme
Z ⊂ X . Following the notation of [10, 11], after picking a curve class β and an integer
n, we denote by
IX (β, n)
the Hilbert scheme of one dimensional subschemes Z ⊂ X satisfying [Z] = β and
χ(OZ ) = n. A simple argument shows that IX (β, n) is the moduli space MX (0, I, β, c3 )
with c3 the third Chern class of any ideal sheaf of Z ⊂ X in IX (β, n). One special feature
of the moduli of rank one torsion free sheaves is that they admit perfect-obstruction
theory for all smooth projective threefolds.
Following [10, 11], for Calabi–Yau threefold X and curve β one forms the generating
function
X
DT X,β (q) =
deg[IX (β, n)]vir qn .
n
In case X is any smooth threefold, since the Hilbert schemes IX (0, n) have virtual
dimensions zero, one defines DT X,β (q) according to the same formula and call it the
dimensional zero Donaldson–Thomas series as well. Of the several conjectures on
DT X,β (q) proposed in [10, 11], one is about the dimension zero Donaldson–Thomas
invariants DT X,0 (q). Let M(q) be the three dimensional partition function
Y
1
M(q) =
(1
−
q n )n
n
and c3 (TX ⊗ KX ) be the third Chern class, viewed as the Chern number, of TX ⊗ KX .
Geometry & Topology 10 (2006)
Zero dimensional Donaldson–Thomas invariants of threefolds
2119
Conjecture 0.1 [10, 11] For any smooth projective threefold X , the dimension zero
Donaldson–Thomas series DT X,0 (q) has the form
DT X,0 (q) = M(−q)c3 (TX ⊗KX ) .
In this paper, we shall prove this conjecture for all compact smooth complex threefolds.
Theorem 0.2 The zero-dimensional Donaldson–Thomas series DT X,0 (q) for any
compact smooth complex threefold X are of the form
DT X,0 (q) = M(−q)c3 (TX ⊗KX ) .
We remark that this conjecture was independently proved for the class of Calabi–Yau
threefolds based on different method by Behrend and Fantechi [3], and for projective
threefolds by Levine and Pandharipande [6].
We now briefly outline the proof of this theorem. Clearly, in case X is a disjoint union
of two smooth proper threefolds X1 and X2 , then
a
IX (0, n) =
IX1 (0, n1 ) × IX2 (0, n2 ).
n1 +n2 =n
Since the deformation of the ideal sheaf of the union Z1 ∪ Z1 ⊂ X1 ∪ X2 is the direct
product of the deformation of Z1 ⊂ X1 and the deformation of Z2 ⊂ X2 , we have
a
[IX (0, n)]vir =
[IX1 (0, n1 )]vir × [IX2 (0, n2 )]vir .
n1 +n2 =n
Therefore
DT X,0 (q) = DT X1 ,0 (q) · DT X2 ,0 (q).
Put differently, the correspondence that sends any threefold to its zero-dimensional
Donaldson–Thomas series defines a homomorphism from the additive semigroup
PC = {All smooth projective threefolds}/iso
to the multiplicative semigroup of infinite series Z[[q]].
A distant cousin of complex manifolds are so called weakly complex manifolds, which
by definition are smooth compact real manifolds M (possibly with smooth boundaries)
together with C–vector bundle structures on the stabilizations1 of their tangent bundles
TM .
Here as usual a stabilization of TM is a direct sum of TM with some trivial bundle Rm on
M ; together with the identity TM ⊕ Rm = (TM ⊕ Rm ) ⊕ R2 with the last R2 is given the obvious
complex structure R2 ∼
= C.
1
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Jun Li
The equivalence classes of weakly complex manifolds (without boundaries) modulo the
relations [∂W] = 0 forms a group, called the complex cobordism group ΩC , under the
`
addition [M1 ] + [M2 ] = [M1 M2 ]. It is a classical result that ΩC ⊗Z Q is generated
by all possible products of projective spaces Pn . As a consequence, the six (real)
dimensional complex cobordism group ΩC
6 is generated by
Y1 = P3 ,
Y2 = P2 ×P1
and Y3 = (P1 )3 .
The crucial step in proving Theorem 0.2 is to establish
Proposition 0.3 There are universal polynomials
f0 , f1 , f2 , · · ·
in Chern numbers of smooth complex threefolds so that for any smooth projective
threefold X its zero-dimensional Donaldson–Thomas series are of the form
X
DT X,0 (q) =
fn (X)qn .
n
Because any complex threefold X is C–cobordant to mm1 Y1 + mm2 Y2 + mm3 Y3 for some
integers ni , knowing the Proposition, and that cobordant weakly complex manifolds
have identical Chern numbers,
X
X
fn (mX)qn =
fn (m1 Y1 + m2 Y2 + m3 Y3 )qn .
n
n
By the definition, the left hand side is
DT mX,0 (q) = DT X,0 (q)m
while the right hand side is
DT m1 Y1 +m2 Y2 +m3 Y3 ,0 (q) = DT Y1 ,0 (q)m1 · DT Y2 ,0 (q)m2 · DT Y3 ,0 (q)m3 .
Because Yi are toric threefolds, their Donaldson–Thomas series are known [10, 11] to
have the form
DT Yi ,0 (q) = M(−q)c3 (TYi ⊗KYi ) .
Put together, and adding that
m c3 (TX ⊗ KX ) =
3
X
mi c3 (TYi ⊗ KYi ),
i=1
we obtain
DT X,0 (q)m = M(−q)
Geometry & Topology 10 (2006)
P3
i=1
mi c3 (TYi ⊗KYi )
= M(−q)m c3 (TX ⊗KX ) .
Zero dimensional Donaldson–Thomas invariants of threefolds
2121
Finally, because both DT X,0 (q) and M(−q) are power series with integer coefficients
and constant coefficient one,
DT X,0 (q) = M(−q)c3 (TX ⊗KX ) .
This would prove the theorem.
As to the proof of the Proposition, we shall first construct a collection of approximations
X [τ ] of X [n] indexed by partitions of [n] = {1, · · · , n}. To each X [τ ] , we shall construct
its virtual cycle and prove that its degree can be approximated by the degree of the
virtual cycle of X [σ] of σ < τ , with errors expressible in terms of a universal expression
of the Chern numbers of X . Thus by induction, we prove that the degree of the virtual
cycle of X [n] can also be expressed universally in terms of the Chern numbers of X ,
thus proving the Proposition and the Theorem.
During the early stage of this work, K Behrend developed a theory of micro-local
analysis for a symmetric obstruction theory [2]; later, jointly with B Fantechi they
proved Conjecture 0.1 for Calabi–Yau threefolds [3]. Toward the end of finalizing this
paper, the author was kindly informed by R Pandharipande that he and M Levine have
proved the same conjecture using algebraic K–theory [6].
The author also like to take this opportunity to thank Weiping Li for his valuable
comments.
0.1
Terminology
We shall work with the category of analytic functions in this paper. Thus for any reduced
quasi-projective scheme W , we shall work with and denote by OW the sheaf of analytic
functions on W ; we shall use ordinary open subsets of the W unless otherwise stated.
To distinguish from that of analytic spaces, we shall reserve the words schemes and
morphisms to mean algebraic schemes and algebraic morphisms, though viewed as
objects in analytic category.
Often, we shall work with open subsets of quasi-projective schemes. We shall endow
such sets with their reduced induced analytic structures and work with their sheaves of
analytic functions. We shall call such space either analytic spaces or analytic schemes.
Accordingly, whenever we say two analytic spaces isomorphic we mean that they are
isomorphic as analytic spaces.
In this paper, we reserve the word smooth to mean the smoothness in C∞ –category.
Thus a smooth function is a C∞ –function and a smooth map is a C∞ –map.
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Jun Li
There is one case in which we need to keep non-reduced scheme structures. It is the
case of flat U –families of zero-subschemes Z ⊂ U × Y with U an analytic space and
Y an open subset of a smooth varieties. In this case, Z is defined by the ideal sheaf
IZ ⊂ OU×Y ; the structure sheaf of Z is the quotient OU×Y /IZ ; we say that Z is flat
over U if OZ is flat over OU .
1
Hilbert schemes of α–points
The purpose of this section is to construct the filtered approximation of IX (0, n) by
Hilbert schemes of α–points.
For convenience, we let
X [n] = IX (0, n)red
be the Hilbert scheme IX (0, n) with the reduced scheme structure.
1.1
The Definition
We begin with a finite set Λ of order |Λ|; it could be the set of n integers [n] = {1, · · · , n}
or a subset of [n]. For such Λ we follow the convention
X Λ = {(xa )a∈Λ | xa ∈ X}.
In case |Λ| = n, we define
X (Λ) = X (n) = Sn X
and
X [Λ] = X [n] .
Using the Hilbert–Chow morphism hc : X [Λ] → X (Λ) , we define
X [[Λ]] = X [Λ] ×X (Λ) X Λ ;
it comes with a tautological projection X [[Λ]] → X Λ . Obviously, according to this
definition a closed point of X [[Λ]] is a pair of a 0–scheme ξ ∈ X [Λ] and a point
P
(xa )a∈Λ ∈ X Λ such that hc(ξ) = xa .
We next look at the set PΛ of all partitions of, or equivalence relations on, the set Λ. In
case α ∈ PΛ has k equivalence classes α1 , · · · , αk , we write
(1)
Geometry & Topology 10 (2006)
α = (α1 , · · · , αk ).
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Zero dimensional Donaldson–Thomas invariants of threefolds
It has two distinguished elements: one is Λ that is the partition with a single equivalence
class Λ; the other is 0Λ that is the partition whose equivalence classes are all single
element sets. The set PΛ has a partial ordering “≤” defined by
“α ≥ β" ⇐⇒ “a ∼β b ⇒ a ∼α b".
In case α = (α1 , · · · , αk ), then α ≥ β if and only if β is finer than α, or that β can be
written as
β = (β11 , · · · , β1l1 , · · · , βk1 , · · · , βklk )
i
so that αi = ∪lj=1
βij .
Under this partial ordering, the element α ∧ β , which is defined by
“a ∼α∧β b" ⇐⇒ “a ∼α b and a ∼β b",
or equivalently for β = (β1 , · · · , βl ) it is α ∧ β = (α1 ∩ β1 , · · · , αk ∩ βl ), is the largest
element among all that are less than or equal to both α and β . Following this rule, 0Λ
is the smallest element and Λ is the largest element in PΛ .
For α ∈ PΛ , we let X [[α]] be (reduced) Hilbert scheme of α–points in X . Let α be as
in (1). Because each αi is a set, we can form X αi , X (αi ) , X [αi ] and X [[αi ]] respectively.
We then define
X (α) =
k
Y
X (αi ) ,
X [α] =
i=1
k
Y
X [αi ] ,
i=1
X [[α]] =
k
Y
X [[αi ]] ;
i=1
they fit into the Cartesian product:
X [[α]] −−−−→ X [α]




y
yhc
S
X Λ −−−α−→ X (α)
We shall call points in X [[α]] α–zero-subschemes and call X [[α]] the Hilbert scheme of
α–points.
The space X [[α]] coincides with X [[β]] over a large open subset of each. For instance, both
X [[0Λ ]] = X Λ and X [[Λ]] contains as their open subsets the set of n = |Λ| distinct ordered
points in X . It is when distinct simple points specialize to points with multiplicities the
space X [[0Λ ]] becomes different from X [[Λ]] : for the former they remain as simple points
by allowing multiple simple points to occupy identical positions in X ; for the later fat
points with non-reduced scheme structures emerge. This way, the collection
{X [[β]] | β ∈ PΛ }
forms an increasingly finer approximation of X [[Λ]] .
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Jun Li
Though the notion of X [[α]] seems artificial at first, it proves to be useful in keeping
tracking of the difference among all X [[α]] . Lastly, in case Λ = [n], we shall follow the
convention
XΛ = Xn,
1.2
X (Λ) = X (n) ,
X [Λ] = X [n] ,
X [[Λ]] = X [[n]] .
The relative case
We can generalize the notion of Hilbert scheme of α–points to that of smooth families
of varieties. Let
π : Y −→ T
be a smooth family of quasi-projective varieties. For any integer l, we let IY/T (0, l) be
the relative Hilbert scheme of length l 0–subschemes of fibers of Y/T . It is the fine
moduli scheme representing the functor parameterizing all flat S–families of length l
0–schemes Z ⊂ Y ×T S. As before, we let
Y [l] = IY/T (0, l)red
be IY/T (0, l) with the reduced scheme structure. The moduli Y [l] is a scheme over T
with a universal family. In case for an open U ⊂ T with Y ×T U = Y0 × U , then
canonically
Y [l] ×T U ≡ Y0[l] × U.
As before, for any α ∈ PΛ we let Y Λ , Y (α) , Y [α] and Y [[α]] be the Cartesian products
presented before with X replaced by Y and with products replaced by fiber products
over the base scheme T . Again in case Y ×T U = Y0 × U ,
Y [[α]] ×T U ≡ Y0[[α]] × U.
The three cases we shall apply this construction is for the trivial fiber bundle pr1 : X ×
X → X , for the total space of the tangent bundle TX → X and for the total space of the
universal quotient bundle Q of a Grassmannian Gr = Gr(N, 3) of quotients C3 of CN .
We shall come back to this in detail later.
1.3
Partial equivalences
As mentioned before, the collection Y [[β]] forms an increasingly finer approximation of
Y [[Λ]] . It is the purpose of this subsection to make this precise.
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Zero dimensional Donaldson–Thomas invariants of threefolds
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We begin with comparing Y [[Λ]] with Y [[α]] for an α = (α1 , · · · , αk ). By definition, a
point in Y [[αi ]] consists of
(ξi , (xa )a∈αi ) ∈ Y [αi ] ×T Y αi
P
subject to the constraint hc(ξi ) = a∈αi xa . In case the support of ξi is disjoint from
that of ξj , then ξi ∪ ξj is naturally a zero-subscheme in Y of length |αi ∪ αj |; the pair
(ξi ∪ ξj , (xa )a∈αi ∪αj ) thus is a point in Y [[αi ∪αj ]] . Applying this to all pairs 1 ≤ i < j ≤ k,
we see that
∪ki=1 ξi , (xa )a∈Λ ∈ Y [[Λ]]
if and only if the supports hc(ξi ) are mutually disjoint.
In general, for α < β = (β1 , · · · , βl ) and 1 ≤ j ≤ l, the pair
∪αi ⊂βj ξi , (xa )a∈βj ∈ Y [[βj ]]
if and only if the supports {hc(ξi ) | αi ⊂ βj } are mutually disjoint. This leads to the
definition
Definition 1.1 For α < β we define ∆(α,β) be the set
{x ∈ Y Λ | xa = xb for at least one pair a, b ∈ Λ so that a ∼β b but a 6∼α b };
for general α 6= β we define ∆(α,β) = ∆(α,α∧β) ∪ ∆(β,α∧β) ; we define the discrepancy
between Y [[α]] and Y [[β]] be
[[α]]
∆(α,β)
, Y [[α]] ×Y Λ ∆(α,β) .
We define
[[α]]
Y(α,β)
= Y [[α]] − ∆[[α]]
(α,β) .
Lemma 1.2 Given any pair α, β ∈ PΛ , we have a functorial isomorphism
[[α]] ∼ [[β]]
Y(α,β)
= Y(β,α) .
Here by functorial isomorphisms we mean those that are induced by the universal
property of the respective moduli spaces.
Proof of Lemma 1.2 We first prove the case α = Λ. Let β = (β1 , · · · , βl ) ∈ PΛ
[[Λ]]
with mi = |βi |; let S = Y(Λ,β)
; and let (W, ϕ) be the tautological family of S. By
definition, ϕ : S → Y Λ is the tautological map and W is a flat S–family of length |Λ|
Geometry & Topology 10 (2006)
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Jun Li
zero-subschemes in Y whose Hilbert–Chow map hcW : S → Y (Λ) coincide with the
composite of ϕ and Y Λ → Y (Λ) .
Similarly, a T –morphism S → Y [[β]] is classified by a collection
(2)
W1 , · · · , Wl ; ϕ1 , · · · , ϕl
of S–families Wi ⊂ Y ×T S in X [βi ] and ϕi : S → Y βi so that the Hilbert–Chow
morphism hcWi : S → X [βi ] (of the family Wi ) makes the diagram
(3)
S


(ϕ1 ,··· ,ϕl )y
(ρW ,··· ,ρW )
1
l
−−−−
−−−−→
X [β]


y
Y Λ −−−−−−−−−→ Y (β)
commutative.
To proceed, we shall show that we can split the family (W, ϕ) into a family as in (2).
For each a ∈ Λ, we denote by ϕa : S → Ya the a-th component of ϕ, by Γϕa ⊂ Y ×T S
[[Λ]]
the graph of ϕa and by Γβi the union ∪a∈βi Γϕa . By the definition of Y(Λ,β)
, the set
Γβ1 , · · · , Γβl are mutually disjoint closed subsets of Y × S; hence are open and closed
subsets of Γ = ∪a∈Λ Γϕa .
On the other hand, the closed subscheme ι : W ,→ Y ×T S is set-theoretically identical
to Γ; hence Wi = ι−1 (Γβi ) are mutually disjoint open and closed subsets of W , which
`
therefore inherit scheme structures from W so that W = li=1 Wi . In particular, they
are close subschemes of Y ×T S, flat and finite over S. Finally, because the fibers of
Wi over general closed points in S have length mi , by the flatness, Wi is a family of
length mi zero-subschemes in Y .
Now, because the sets Γβi are mutually disjoint, the associated classifying morphisms
Y
ρWi : S −→ Y [βi ] and ϕi =
ϕa : S −→ Y βi
a∈βi
satisfies the commutative diagram (3). Therefore, each pair (Wi , ϕi ) associates to a
canonical morphism
ηi : S −→ Y [βi ] ×Y (βi ) Y βi = Y [[βi ]] .
Put them together, we obtain a morphism
η = (η1 , · · · , ηl ) : S −→ Y [[β1 ]] ×T · · · ×T Y [[βl ]] = Y [[β]]
[[β]]
whose image is contained in Y(β,Λ)
. This way, we have constructed an induced morphism
[[β]]
[[Λ]]
η : S = Y(Λ,β)
−→ Y(β,Λ)
.
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Zero dimensional Donaldson–Thomas invariants of threefolds
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To complete the proof of this special case, we need to construct a morphism
[[β]]
[[Λ]]
η : Y(β,Λ)
−→ S = Y(Λ,β)
that is the inverse of η . But this is straight forward and shall be omitted. This proves
the lemma for the case α = Λ.
For the general case α = (α1 , · · · , αk ) ∈ PΛ , the α ∧ β consists of equivalence classes
αi ∩ βj , each of order mij . Obviously,
αi ∧ β , (αi ∧ β1 , · · · , αi ∧ βl ) ∈ Pαi .
Hence we can apply the proven case of this lemma to conclude
[[αi ∧β]] ∼ [[αi ]]
Y(α
= Y(αi ,αi ∧β) .
i ∧β,αi )
Therefore, because Y [[β]] =
[[α]]
Y(α,α∧β)
=
=
Qk
T
k
Y
Y [[βi ]] , one checks easily that:
[[αi ]]
Y(α
i ,αi ∧β)
i=1
k Y
l
Y
∼
=
k
Y
[[αi ∧β]]
Y(α
i ∧β,αi )
i=1
Y [[αi ∧βj ]] ×Y αi Y αi − ∆(αi ,αi ∧β)
i=1 j=1
=
Y
Y [[αi ∧βj ]] ×Y Λ Y Λ − ∆(α,α∧β)
i,j
For the same reason,
[[β]]
∼
Y(β,α∧β)
=
Y
Y [[αi ∧βj ]] ×Y Λ Y Λ − ∆(β,α∧β) .
i,j
Because ∆(α,β) = ∆(α,α∧β) ∪ ∆(β,α∧β) , we obtain
[[β]]
[[β]]
[[α]]
[[α]]
Y(α,β)
= Y(α,α∧β)
×Y Λ Y Λ − ∆(α,β) ∼
= Y(β,α∧β) ×Y Λ Y Λ − ∆(α,β) = Y(α,β) .
This proves the Lemma.
1.4
Universal families under partial equivalence
Let (α, β) and α ∧ β = (α1 ∩ β1 , · · · , αk ∩ βl ) be as in the proof of the previous lemma;
let
(Wij , ϕij : i = 1, · · · , k, j = 1, · · · , l)
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Jun Li
[[α∧β]]
[[α]]
∼
be the universal family of Y [[α∧β]] . Under the partial equivalence Y(α,α∧β)
= Y(α∧β,α) ,
[[α∧β]]
the restriction of Wij to Y(α∧β,α)
:
[[α∧β]]
[[α∧β]]
Wi1 ×Y [[α∧β]] Y(α∧β,α)
, · · · , Wil ×Y [[α∧β]] Y(α∧β,α)
,
[[α]]
are families of zero-schemes of Y over Y(α∧β,α)
. Following the previous proof, these
[[α∧β]]
families, viewed as subschemes in Y ×T Y(α∧β,α)
, are mutually disjoint. Hence their
union
l
a
[[α∧β]]
Wij ×Y [[α∧β]] Y(α∧β,α)
j=1
[[α∧β]]
forms a flat family of zero schemes in Y of length |αi | over Y(α∧β,α)
.
Corollary 1.3 Let (Zi , ϕi ; i = 1, · · · , k) be the universal family of Y [[α]] . Then for
each i,
l
a
[[α∧β]]
[[α]]
Wij ×Y [[α∧β]] Y(α∧β,α)
== Zi ×Y [[α]] Y(α,α∧β)
j=1
as families of relative zero-subschemes in Y/T .
1.5
Hilbert scheme of centered α-points
In order to parameterize family of slices in Y [[α]] , we need the notion of Hilbert scheme
of centered α–points.
Let π : Y → T be the total space of a rank three vector bundle, viewed as a smooth
family of affine schemes isomorphic to A3 . We let
1 X
$ : Y Λ −→ V, (xa )a∈Λ ∈ YtΛ 7−→
xa ∈ Yt
|Λ| a
be the fiberwise averaging morphism and let
$[[α]] : Y [[α]] −−→ Y
be its composition with the tautological Y [[α]] → Y Λ . We define the relative Hilbert
scheme of centered α–points be the preimage of the zero section 0Y of Y under $[[α]] :
(4)
Y0[[α]] = Y [[α]] ×Y 0Y .
Intuitively, Y0[[α]] consists of α–zero-subschemes whose center of support lie in the zero
section of Y .
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For any pair α, β ∈ PΛ , we define
[[α]]
[[α]]
= Y0[[α]] ∩ Y(α,β)
.
Y0,(α,β)
The partial equivalence for Y [[α]] carries over to
[[β]]
[[α]]
= Y0,(β,α)
Y0,(α,β)
(5)
1.6
Outline of the proof
We now explain briefly the strategy to prove the main Proposition. First, because X [[n]]
is finite over X [n] , the degree of its virtual cycle is a fraction of that of X [n] . To study
the former, we first make sense of the virtual cycles [X [[α]] ]vir for all partitions α of [n];
we then construct explicitly their cycle representatives Dα as cycles in X n . Using the
[[α]] ∼ [[β]]
isomorphism X(α,β)
= X(β,α) , we can choose Dα and Dβ so that their difference lies
entirely in a small tubular neighborhood of ∆(α,β) ⊂ X n . Repeating this procedure,
we show that the desired degree deg D[n] is a linear combination of deg Dα plus a
discrepancy term which we denote by δ[n] . Using induction on n, to prove the main
proposition we only need to show that δ[n] only depend on the Chern numbers of X .
To prove the last statement, we shall find a cycle representative of δ[n] that is entirely
contained in a small (tubular) neighborhood of the top diagonal
Λ
X∆
= {(x, · · · , x) | x ∈ X} ⊂ X n .
Once we know this, we shall find a small (tubular) neighborhood U ⊂ X [[Λ]] of
[[n]]
n
X∆
= X [[n]] ×X n X∆
(6)
and a fibration
π : U −→ X
whose homotopy type is determined by one of its fiber π −1 (x) and the tangent bundle
TX . Since the homotopy type of the fiber π −1 (x) is universal (independent of threefold
X ), the discrepancy δ[n] thus only depend on the homotopy type of TX . This will lead
to a proof that δ[n] , and thus deg[X [[n]] ]vir , depends only on a universal expression in
Chern numbers of X .
2
Top diagonal of the Hilbert scheme of α–points
In this section, we shall give a smooth parameterization of the normal slices to the
top diagonal in the Hilbert scheme X [[α]] . Before doing this, we shall comment on the
terminology on stratified spaces and their smooth functions.
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2.1
Jun Li
Stratifications of singular spaces
Since we primarily are interested in (reduced) quasi-projective schemes and their open
subsets, we shall confine ourselves to their stratifications and functions.
For quasi-projective W , we shall only consider stratifications by Zariski locally closed
smooth subvarieties. It is known that every quasi-projective scheme admits such
stratifications. In case we are given a finite collection R of Zariski closed subsets
of W , we can find stratifications S of W subordinating to R in the sense that each
R ∈ R is a union of strata in S . To find a canonical such stratification, we can take the
smallest such stratification2 among those subordinating to R. In case in addition we
are given a morphism of schemes π : C → W , we can find a stratification S 0 of C and
a stratification S of W so that S is subordinating to R and π : C → W is a stratified
map. Again, among all such pairs of stratifications there is one that is the smallest; we
call such pair the standard stratification of C → W subordinating to R.
The collection R usually arises from the singular loci of a sheaf E of OW –modules.
To such sheaf E we associate a collection inductively by letting R0 = W and letting
Ri ⊂ Ri−1 be the non-locally free locus of the sheaf of ORi−1 –modules E ×OW ORi−1 .
We shall call {Ri } the loci of non-locally freeness of E .
We next look at the standard stratification of Y [[α]] for a smooth family over a smooth T .
Obviously, in case U ⊂ T is a Zariski open so that Y ×T U = Y0 × U , then strata of
the standard stratification of Y [[α]] ×T U are of the form S × U for S strata of Y0[[α]] . In
case Y/T is the total space of a rank three vector bundle, then Y0 = A3 and we know
that all strata of Y0[[α]] are invariant under the symmetry group of Y0 . This proves
Lemma 2.1 Let Y/T be a smooth Zariski fiber bundle. Then the standard stratification
of Y [[α]] restricts to the standard stratification of (Yt )[[α]] = (Y [[α]] )t .
Another property of the standard stratification of X [[α]] is as follows.
Lemma 2.2 Let Ua ⊂ X and Ub ⊂ X be two analytic open subset that are biholomorphic to each other, say via f : Ua −→ Ub . Then pulling back via f ∗ of the universal
family I of X [[α]] over Ub[[α]] , X [[α]] ×X Λ UaΛ defines a map
F : Ub[[α]] −→ Ua[[α]]
that is an isomorphism of (standardly) stratified spaces. This construction also applies
to smooth family of quasi-projective varieties over a smooth base.
2
Two stratifications S1 ≤ S2 if each strata in S1 is the union of strata in S2 .
Geometry & Topology 10 (2006)
Zero dimensional Donaldson–Thomas invariants of threefolds
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Here we say a continuous map f : Z1 → Z2 between two stratified spaces preserves
stratifications if every stratum of Z1 is the preimage of a stratum of Z2 . We say f is a
smooth isomorphism of stratified spaces if f −1 exists and both f and f −1 are smooth
and preserve stratifications of Z1 and Z2 .
Proof Let α = (α1 , · · · , αk ). Because Ua[[α]] is canonically isomorphic to
Y
U αi ×U (α) Ua[αi ] ,
a
to prove the lemma it suffices to show that for every n the Hilbert scheme of n–points
IUa (0, n) ∼
= IUb (0, n) canonically under the map induced by f . But this follows from the
universal property of Hilbert schemes.
2.2
Smooth functions on stratified spaces
We now define the notion of smooth functions on a stratified space.
Definition 2.3 Let Z be a topological space equipped with a stratification S . A
smooth function f : Z → C is a continuous function whose restriction to each stratum
is smooth.
We let W be an analytic space and let E be a sheaf of OW –modules. We let S be a
standard stratification subordinating to the loci of the non-locally freeness of E . To
such stratification, we denote by AS the sheaf of smooth functions on W , and define
the sheaf of smooth sections of E be
AS (E) = E ⊗OW AS .
Because AS has partition of unity, whenever 0 → E2 → E → E1 → 0 is an exact
sequence of sheaves of OW –modules, then both
AS (E2 ) −→ AS (E) −→ AS (E1 ) −→ 0
and
Γ AS (E2 ) −→ Γ AS (E) −→ Γ AS (E1 ) −→ 0
are exact.
As usual, for any subset B ⊂ W and sheaf of OW –modules E , the space of smooth
sections of E over B is the limit
Γ(B, AS (E)) = lim Γ(U, AS (E))
B⊂U
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Jun Li
taken over all open subsets U containing B. Thus each s in Γ(B, AS (E) is a smooth
section defined on some open U containing B. Further, in case s̃ ∈ Γ(B0 , AS (E)) is a
section over a larger subset B0 ⊃ B, we say s0 extends s if there is an open U ⊃ B so
that both s and s0 are defined over U and s|U ≡ s0 |U . Following this convention, the
restriction homomorphism
Γ(B0 , AS (E)) → Γ(B, AS (E))
is surjective for any pair of sets B ⊂ B0 with B closed in B0 ,
2.3
Normal slices to the diagonal in X × X
We begin with constructing an isomorphism of a tubular neighborhood U of ∆(X) ⊂
X × X with a tubular neighborhood V of the zero section 0X ⊂ TX . This map will be a
smooth isomorphism of analytic fiber bundle U/X and V/X in the sense that for each
x ∈ X the fiber Ux and Vx are isomorphic as analytic spaces. Here U/X is via the first
projection pr1 : U ⊂ X × X −→ X .
Since Vx ⊂ Tx X , it has a distinguished point 0 ∈ Vx and T0 Vx ≡ Tx X canonically;
likewise, Ux is a neighborhood of x ∈ X , which has its distinguished point x ∈ Ux and
isomorphism Tx Ux ≡ Tx X .
Lemma 2.4 We can find a smooth isomorphism
ϕ : U −−→ V
of a tubular neighborhood U of ∆(X) ⊂ X × X and a tubular neighborhood V of
0X ⊂ TX , both considered as vector bundles over X , such that
(1) restricting to each fiber Ux the map ϕx = ϕ|Ux : Ux → Vx is a biholomorphism,
(2) ϕx (x) = 0 and that dϕx : Tx Ux → T0 Vx is the identity map.
Proof First, after identifying Tx V ∼
= V in the standard way for the threefold V = A3
we can define such ϕ globally:
φ : V × V −→ TV;
V × V 3 (x, v) 7−→ v − x ∈ Tx V.
Hence for any open Ua ⊂ X that admits an open embedding Ua ⊂ V , the map φ
restricting to
Uα = φ−1 (TUa V) ∩ (Ua × Ua ) ⊂ X × X
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is a local version of the map required by the lemma; we denote such φ|Uα by
ϕa : Ua ⊂ X × X −→ TX.
(7)
We next pick an open covering {Ua } of X with embedding Ua ⊂ V and a partition of
P
unity
ηa ≡ 1 subordinate to the covering Ua ; we form
U , {(x, y) ∈ X × X | (x, y) ∈ Ua whenever x ∈ Supp(ηα )}.
Because Supp(ηα ), which is the closure of ηα =
6 0, is compact and is contained in Ua ,
U is open and contains the diagonal ∆(X) ⊂ X × X . Over U , we define
X
ϕ(x, y) =
ηa (x)ϕa (x, y) ∈ TX.
α
We let π : U → X be the projection induced by the second projection pr2 : X × X → X .
Clearly, restricting to each fiber Ux = π −1 (x),
ϕx (·) = ϕ(x, ·) : Ux → Tx X
is analytic. Further, because ϕa (x, x) = 0 and dy ϕa (x, y)|y=x = id whenever ηa (x) 6= 0,
we have ϕx (x) = 0 and dϕx : Tx Ux → T0 Vx is the identity map. Therefore, by replacing
U by a sufficiently small tubular neighborhood U 0 of ∆(X) ⊂ X × X and let V 0 = ϕ(U 0 ),
the restriction ϕ0 , ϕ|U 0 becomes the desired map.
2.4
Induced map on Hilbert schemes
The map ϕ induces a smooth map from the relative Hilbert scheme U [[α]] to V [[α]] as
fiber bundles over X .
We begin with the definition of U [[α]] and V [[α]] . First, by letting Y = TX be the vector
bundle over T = X , we can apply the previous discussion to form the relative Hilbert
scheme of α–points (TX)[[α]] . Let (TX)[[α]] → (TX)Λ be the tautological map; let V Λ
be the product of |Λ|–copies of V over X indexed by Λ, which is an open subset of
(TX)Λ . We define V [[α]] , called the relative Hilbert scheme of α–points in V/X , be an
open subset of (TX)[[α]] defined via the Cartesian product (to the left below)
(8)
V [[α]] −−−−→ (TX)[[α]]




y
y

V Λ −−−Λ−→ (TX)Λ
U [[α]] −−−−→ X × X [[α]]




y
y
ι
U Λ −−−Λ−→ X × X Λ
Similarly we form the relative Hilbert scheme U [[α]] of α–points of U/X as an open
subset of X × X [[α]] defined by the square (to the right above). Both U [[α]] and U [α] are
analytic spaces over X via the first projection of X × X [[α]] and X × X Λ .
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Jun Li
Because for each closed x ∈ X , the fiber ϕx : Ux → Vx is an isomorphism, restricting to
fibers over x the universal families of U [[α]] and V [[α]] induces a canonical isomorphism
[[α]] −→ V [[α]] from U [[α]] = U [[α]] × x to V [[α]] = V [[α]] × x. We define
ϕ[[α]]
X
X
x : Ux
x
x
x
ϕ[[α]] : U [[α]] −→ V [[α]]
(9)
be ϕ[[α]]
when restricting to Ux[[α]] .
x
Lemma 2.5 The map ϕ[[α]] is a smooth isomorphism of the analytic fiber bundles
U [[α]] and V [[α]] as stratified spaces. Namely, ϕ[[α]] preserves the standard stratifications
of V [[α]] and U [[α]] and
(ϕ[[α]] )−1 AV [[α]] = AU [[α]] .
Further, restricting to fibers over each x, ϕ[[α]] induces isomorphism of analytic spaces
Ux[[α]] and Vx[[α]] .
Proof The proof is a tautology after applying the universal property of the moduli
spaces U [[α]] and V [[α]] . First, because ϕ : U → V is a smooth isomorphism that
preserves the analytic structures of the fibers, pulling back the universal family of V [[α]]
via ϕ forms a continuous family of relative α–schemes of the fiber bundle U/X , thus
defines a continuous map
ψ [[α]] : V [[α]] −→ U [[α]] .
Now, let S ⊂ V [[α]] be any stratum of the standard stratification of V [[α]] . Because its
(standard) stratification is induced from that of (TX)[[α]] , the stratum S is the restriction
of a stratum S̃ of (TX)[[α]] . By Lemma 2.1, for an x ∈ T the intersection S̃ ∩ (TX)[[α]]
x
is a stratum of (Tx X)[[α]] . Hence Sx = S ∩ Vx[[α]] is a stratum of Vx[[α]] . Then because
ψx[[α]] : Vx[[α]] → Ux[[α]] is an analytic isomorphism, Sx0 = ψx[[α]] (Sx ) must be a stratum of
Ux[[α]] . Similar to the case TX/X , Sx0 is the intersection with Ux[[α]] of a single stratum S̃0
of X × X [[α]] . Let S0 = S̃0 ∩ U [[α]] . Applying Lemma 2.2, we see immediately that
ψ [[α]] (S) = S0 .
Thus ψ [[α]] preserves the stratifications.
On top of this, because restricting to S the pull back of the universal family is a smooth
family of relative α–subschemes, ψ [[α]] |S is a smooth map from S to ψ [[α]] (S). This
proves that ψ [[α]] is a smooth stratified map.
Similarly, applying the same argument to ϕ−1 : V → U , we obtain the map ϕ[[α]] that
is also a smooth stratified map. Then the composition
ψ [[α]] ◦ ϕ[[α]] : U [[α]] −→ U [[α]]
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Zero dimensional Donaldson–Thomas invariants of threefolds
must be the identity map since it is induced by the identity ϕ ◦ ϕ−1 = id. For the same
reason, ϕ[[α]] ◦ ψ [[α]] = id as well. This proves that both ϕ[[α]] and ψ [[α]] are smooth
isomorphisms of stratified spaces. Lastly, because ψ [[α]] preserves fibers over x, it
defines a smooth map from V [[α]] to U [[α]] as smooth fiber bundles over X .
The last statement is true because restricting to fibers over x ∈ X , both ϕ[[α]] and ψ [[α]]
are analytic isomorphisms. This proves the Lemma.
2.5
Top diagonal and its normal slices
[[α]]
We define the top diagonal X∆
⊂ X [[α]] be
[[α]]
Λ
X∆
= X∆
×X Λ X [[α]]
Λ
where X∆
= {(x, · · · , x) ∈ X Λ | x ∈ X}.
Λ ∼ X , the top diagonal is fibred over X :
Because X∆
=
[[α]]
X∆
−−→ X.
(10)
[[α]]
⊂ X [[α]] and
The purpose of this subsection is to find an open neighborhood U of X∆
a fiber bundle map U → X extending the map (10).
We let (TX)[[α]]
be the relative Hilbert scheme of centered α–points of the fiber bundle
0
TX/X . Because V [[α]] is an open subbundle of (TX)[[α]] , we define the relative Hilbert
scheme of centered α–points of V/X be
V0[[α]] = V [[α]] ∩ (TX)0[[α]] .
Λ
Λ
There is another way to define this space. Let (TX)Λ
0 = (TX) ×TX 0TX and let V0 be
[[α]]
the intersection V Λ ∩ (TX)Λ
and V0Λ fit into the following commutative
0 . Then V0
Cartesian squares
ψ [[α]]
(11)
[[α]]
pr
1
V0[[α]] −−−−→ V [[α]] −−−−→ U [[α]] −−−−→ X [[α]] × X −−−−
→ X [[α]]










y
y
y
y
y
ψ

pr
1
V0Λ −−−−→ V Λ −−−Λ−→ U Λ −−−Λ−→ X Λ × X −−−−
→ XΛ
in which the ψΛ , Λ and [[α]] are the obvious maps induced by ψ and by  : U → X ×X .
We let Ψα : V0[[α]] → X [[α]] be the composite of the arrows in the top line.
Lemma 2.6 After shrinking V if necessary, the image set
U [[α]] , Ψα V0[[α]] ⊂ X [[α]]
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Jun Li
[[α]]
is an open neighborhood of X∆
⊂ X [[α]] . Further, after endowing U with the induced
stratification through the inclusion U [[α]] ⊂ X [[α]] , the induced map
Ψα : V0[[α]] −−→ U [[α]]
becomes a smooth isomorphism of stratified spaces.
Proof Because each square in the above diagram is a Cartesian product square, to
prove that Ψα V0[[α]] is open in X [[α]] it suffices to show that the image of V0Λ in X Λ
under the composite of the bottom line is open in X Λ .
We let h : V0Λ → X Λ be the composite of the bottom line in the above diagram. We
first prove that the differential
dh(ξ) : Tξ V0Λ −−→ Th(ξ) X Λ
is an isomorphism at each ξ in the zero section 0(TX)Λ ⊂ (TX)Λ .
Let ξ ∈ 0(TX)Λ be a point over x ∈ X . Since (Tx X)Λ
0 intersects 0(TX)Λ transversal at ξ ,
Λ
Λ
the tangent space Tξ V0 is a direct sum of Tξ (Tx X)0 and Tξ 0(TX)Λ . Clearly, the images
d h Tξ (Tx X)Λ
0 and d h Tξ 0(TX)Λ are
X
{(va )a∈PΛ ∈ (Tx X)Λ |
va = 0} and {(v, · · · , v) ∈ (Tx X)Λ | v ∈ Tx X}
respectively. Thus d h is an isomorphism at ξ ∈ 0(TX)Λ ⊂ V0Λ and thus h is a
diffeomorphism near 0(TX)Λ ⊂ V0Λ . In particular, if we shrink V if necessary, h
becomes a diffeomorphism from V0Λ to the image h V0Λ , an open neighborhood of
Λ ⊂ XΛ .
X∆
[[α]]
It follows then that U [[α]] = Ψα V0[[α]] is an open neighborhood of X∆
⊂ X [[α]] and the
[[α]]
induced map Ψα is continuous and is one–one and onto. We endow U
= Ψα V0[[α]]
the stratification induced from that of X [[α]] .
We now prove that Ψα is a smooth isomorphism of stratified spaces. We first prove
that Ψα preserves the stratifications. Indeed, because all but the furthest left arrow
in the top line of the above diagram preserves stratifications, to prove the claim we
only need to show that the inclusion V0[[α]] ⊂ V [[α]] preserves stratifications, which
follows from that the inclusion (TX)[[α]]
→ (TX)[[α]] preserves stratifications. For the
0
[[α]]
[[α]]
×TX 0(TX)Λ with $ : (TX)[[α]] → TX the fiberwise
later, because (TX)0 = (TX)
averaging morphism, we only need to prove that the restriction of $ to each stratum
S ⊂ (TX)[[α]]
$|S : S −−→ TX
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is a submersion. But this is clear since the stratum S is induced by a stratum S0 of
(Tx X)[[α]] , as shown in Lemma 2.1, and the stratum S0 is invariant under the translation
group of Tx X . Therefore $|S is a submersion.
Once we know that Ψα : V0[[α]] → X [[α]] preserves stratifications, the fact that each
squares above is a Cartesian product shows immediately that it is smooth, and its
restriction to each stratum is a diffeomorphism onto its image. Hence, Ψα is a smooth
isomorphism of stratified spaces.
⊂ (TX)[[α]] is a subscheme, hence V0[[α]] ⊂ V [[α]] is an
From the definition, (TX)[[α]]
0
[[α]]
analytic subscheme. Therefore, restricting to fibers U0,x
of U0[[α]] over x ∈ X the map
Φ is analytic.
Before we close this subsection, we shall comment on the partial equivalence of V0[[α]]
and of U [[α]] . For this, we define
[[α]]
V0,(α,β)
= V0[[α]] ∩ (TX)[[α]]
0,(α,β)
and
[[α]]
[[α]]
U(α,β)
= U [[α]] ∩ X(α,β)
.
Lemma 2.7 Let Ψα : V0[[α]] → U [[α]] be the smooth isomorphism constructed before.
[[α]] [[α]]
Then Ψα V0,(α,β)
= U(α,β)
and the partial equivalence of V0[[α]] and of U [[α]] are
compatible under Ψα .
Proof This is obvious and will be omitted.
2.6
Universal family of Hilbert schemes of centered α–points
We let Q → Gr be the total space of the universal quotient rank three vector bundle
over the Grassmannian Gr = Gr(CN , C3 ) and let ρ : Q[[α]]
→ Gr be the associated
0
relative Hilbert scheme of centered α–points. Let
g : X −−→ Gr
be a smooth map so that TX ∼
= g∗ Q as smooth vector bundles. To such g, we define the
pull back
a
[[α]]
g∗ Q[[α]]
=
Q
×
X
=
ρ−1 (g(x)) ⊂ Q[[α]]
× X.
Gr
0
0
0
x∈X
Q[[α]]
0
Further, the universal family of
pulls back under g to a continuous family of
relative centered α–points of TX/X , thus defines a continuous map over X
(12)
gα : g∗ Q[[α]]
−−→ (TX)[[α]]
0
0 .
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Lemma 2.8 This map is a smooth isomorphism of fiber bundles over X . Further, its
restriction to each fiber is an analytic isomorphism and preserves the partial equivalences
of Q[[α]]
and of (TX)[[α]]
0
0 .
3
Obstruction sheaves
Our next step is to investigate the obstruction theory of the relative Hilbert scheme of
α–points of a smooth family Y/T . Because Y Λ → Y (α) is not étale, the defining square
Y [[α]] −−−−→


y
Y [α]


y
Y Λ −−−−→ Y (α)
does not allow us to lift the obstruction theory of Y [α] to that of Y [[α]] . Our solution
is to take the pull-back of the obstruction sheaf and the normal cone of Y [α] as the
obstruction sheaf and normal cone of Y [[α]] . To force Y [[α]] → Y [α] flat, we shall view
Y Λ → Y (α) , and hence Y [[α]] → Y [α] , as a morphism between stacks.
We will investigate the obstruction sheaves in this section, deferring normal cones to
the next section. We begin with a brief account of the relevant extension sheaves.
3.1
A brief account of extensions sheaves
Let Y/T be a smooth family of quasi-projective threefolds over a smooth variety T ; let
S be any scheme over T and Z be a flat S–family of 0–subschemes in YS = Y ×T S; and
let pS : YS → S be the projection. Our first goal is to prove a canonical isomorphism
relating the traceless part of the relative extension sheaf of the ideal sheaf IZ of Z ⊂ YS
with the high direct image sheaf of an extension sheaf.
Lemma 3.1 Let the notation be as before, then we have canonical isomorphism
Ext 2pS IZ , IZ 0 ∼
= pS∗ Ext 3OY (OZ , IZ ).
S
Proof We begin with the defining exact sequence
0 −→ IZ −→ OYS −→ OZ −→ 0
and its associated long exact sequence of relative extension sheaves
φ2
−→ Ext 2pS (OYS , IZ ) −→ Ext 2pS (IZ , IZ ) −→ Ext 3pS (OZ , IZ ) −→
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Zero dimensional Donaldson–Thomas invariants of threefolds
φ3
−→ Ext 3pS (OYS , IZ ) −→ Ext 3pS (IZ , IZ ) −→ 0.
(13)
Because OYS is locally free and Z is a flat family of zero-subschemes YS ,
Ext kpS (OYS , IZ ) = Rk pS∗ IZ = Rk pS∗ OYS ,
k ≥ 2.
On the other hand, it follows from the definition that Rk pS∗ OYS is a subsheaf of
Ext kpS (IZ , IZ ) and the composite
(14)
⊂
tr
Rk pS∗ OYS −→ Ext kpS (IZ , IZ ) −→ Rk pS∗ OYS
is multiplying by 1, which is the rank of IZ . Thus Rk pS∗ OYS is the trace part of
Ext kpS (IZ , IZ ), and the traceless part
(15)
Ext 2pS (IZ , IZ )0 ∼
= Ext 3pS (OZ , IZ )
canonically.
To complete the proof of the lemma, we shall apply the local to global spectral sequence.
First because the extension sheaf Ext i (OZ , IZ ) is zero away from Z and Z → S has
relative dimension 0,
Rk pS∗ Ext 3−k (OZ , IZ ) = 0
except when k = 0. Hence the spectral sequence
R• pS∗ Ext • (OZ , IZ ) ⇒ Ext •pS (OZ , IZ )
degenerates to
Ext 3pS OZ , IZ
0
∼
= pS∗ Ext 3 (OZ , IZ ).
This proves the Lemma.
The relative extension sheaves satisfy the base change property. Let ρ : S0 → S be a
morphism and Z 0 = Z ×S S0 be the subscheme of YS0 = YS ×S S0 , then
(16)
Ext 2pS0 IZ 0 , IZ 0 0 ∼
= ρ∗ Ext 2pS IZ , IZ 0
and such isomorphisms are compatible with S00 → S0 → S.
To prove the base change property, we will apply the Lemma and prove the base change
property of
pS∗ Ext 3 (OZ , IZ ).
Because dim Y/T = 3 and OZ is flat over S, the later admits a length three locally free
resolution as a sheaf of OYS –modules. Then because tensoring this resolution by OS0
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Jun Li
provides a locally free resolution of OZ 0 , applying the definition of extension sheaves
we readily see that
Ext 3OY 0 (OZ 0 , IZ 0 ) ∼
= Ext 3OY (OZ , IZ ) ⊗OS OS0 .
(17)
S
S
Because the first sheaf if a finite sheaf of OS0 –modules,
pS0 ∗ Ext 3OY 0 (OZ 0 , IZ 0 ) = Ext 3OY 0 (OZ 0 , IZ 0 )
S
S
when the later is viewed as sheaf of OS0 –modules. Thus by viewing the other Ext 3
sheaf as a sheaf of OS –modules, the identity is exactly the base change property (16).
Corollary 3.2 Let Z ⊂ YS be as in the previous lemma. Suppose further that
Z = Z1 ∪ Z2 is a union of two disjoint flat S–families of 0–subschemes. Then
Ext 2pS IZ , IZ 0 ∼
= Ext 2pS IZ1 , IZ1 0 ⊕ Ext 2pS IZ2 , IZ2 0
canonically.
Proof This is true because
Ext 2pS IZ , IZ
0
∼
= pS∗ Ext 3 (OZ , IZ ) ∼
=
2
M
pS∗ Ext 3 (OZi , IZi ).
i=1
3.2
Obstruction sheaves of Hilbert scheme of points
For a smooth family of quasi-projective threefolds over a smooth base T , the relative
Hilbert scheme IY/T (0, n) is a quasi-project fine moduli scheme with universal family
Z ⊂ Y ×T IY/T (0, n);
its obstruction theory as moduli of stable sheaves with fixed determinants is perfect
with obstruction sheaf the traceless part of the relative extension sheaf under the second
projection of Y ×T IY/T (0, n):
ObIY/T (0,n) = Ext 2pr2 (IZ , IZ )0 .
For Y [n] , which is IY/T (0, n) endowed with the reduced scheme structure, we shall take
the pull back of the obstruction sheaf of IY/T (0, n) under the inclusion Y [n] → IY/T (0, n)
as its obstruction sheaf. By the base change property just proved, it can also be expressed
as the traceless part of the relative extension sheaf of the universal ideal sheaf of Y [n] .
Now let α = (α1 , · · · , αk ) be any element in PΛ as before; let (Z1 , · · · , Zk ) be the
universal family of Y [α] , each is a flat family of length |αi | zero-subschemes in Y/T
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Zero dimensional Donaldson–Thomas invariants of threefolds
Q
over Y [α] ; we let IZi be the ideal sheaf of Zi ⊂ Y ×T Y [α] . Because Y [α] = T Y [αi ] ,
we define the obstruction sheaf of Y [α] be the direct sum of the pull back of that of
Y [αi ] , which by Corollary 3.2 is of the form
Ob[α]
Y =
k
M
Ext 2pr2 IZi , IZi 0 .
i=1
As to Y [[α]] , we shall take the pull-back φ∗α Ob[α] (of the tautological φα : Y [[α]] → Y [α] )
as its obstruction sheaf. For the same reason, they can be defined using relative extension
sheaves. Let (W1 , · · · , Wk ) with Wi ⊂ Y ×T Y [[α]] be part of the universal family of
Y [[α]] ; each Wi is the pull back of Zi under Y [[α]] → Y [α] ; let IWi be the ideal sheaf of
Wi ⊂ Y ×T Y [[α]] . The obstruction sheaf of Y [[α]] is
ObY[[α]]
≡
k
M
Ext 2pr2 (IWi , IWi )0 .
i=1
3.3
Comparing obstruction sheaves under equivalences
Our next task is to compare the sheaves Ob[[α]]
with Ob[[β]]
over the partial equivalence
Y
Y
[[α]] ∼ [[β]]
Y(α,β) = Y(β,α) .
[[α]] ∼ [[β]]
[[α]]
Lemma 3.3 Under the partial equivalence Y(α,β)
= Y(β,α) , the restriction to Y(α,β) of
[[β]]
the obstruction sheaf Ob[[α]]
is canonically isomorphic to the restriction to Y(β,α)
of
Y
Ob[[β]]
Y .
Proof We first show that the Lemma can be reduced to the case where α ≥ β . Indeed,
[[α∧β]]
[[α]] ∼ [[β]]
[[α]]
[[α]]
because Y(α,β)
= Y(β,α) is induced by Y(α,β) ⊂ Y(α,α∧β) ∼
= Y(α∧β,α) , should the lemma
hold for α ≥ β , we would have
ObY[[α]] |Y [[α]]
(α,α∧β)
[[α∧β]]
∼
|Y [[α∧β]] ,
= ObY
(α∧β,α)
which would imply
[[α∧β]]
∼
Ob[[α]]
|Y [[α]]
Y |Y [[α]] = ObY
(α,β)
[[β]]
(α,β) =Y(β,α)
[[β]]
∼
= ObY |Y [[β]] .
(β,α)
For the case α ≥ β , by induction we only need to consider the case where
α = (α1 , · · · , αk ) ≥ β = (β1 , · · · , βk+1 ).
For simplicity we assume αi = βi for i < k and αk = βk ∪ βk+1 .
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Jun Li
[[α]]
Now let (Wi , ϕi )1≤i≤k , with Wi ⊂ Y ×T Y(α,β)
be the universal family of Y [[α]] over
[[α]]
[[β]] over Y [[β]] . Because α = β for
Y(α,β)
; let (W̃i , ϕ̃i )k+1
i
i
i=1 be the universal family of Y
(β,α)
[[α]] ∼ [[β]]
∼
i < k, Wi = W̃i after identifying Y
. On the other hand, by Corollary 1.3
=Y
(α,β)
(β,α)
[[β]]
the supports of W̃k and W̃k+1 are disjoint closed subsets of Y ×T Y(β,α)
; and their
union form a flat family of zero-subschemes satisfying W̃k ∪ W̃k+1 = Wk .
As for the obstruction sheaves Ob[[α]]
and Ob[[β]]
Y
Y , by definition,
ObY[[α]] |Y [[α]] =
(α,β)
Ob[[β]]
Y |Y [[β]] =
and
(β,α)
k
M
i=1
k+1
M
Ext 2pr2 IWi , IWi
0
Ext 2pr2 IW̃i , IW̃i 0 .
i=1
Hence to prove the lemma we only need to check that
Ext 2pr2 IWk , IWk 0 ∼
= Ext 2pr2 IW̃k , IW̃k 0 ⊕ Ext 2pr2 IW̃k+1 , IW̃k+1 0 .
But this is exactly what was proved in the last subsection.
3.4
Obstruction sheaves under smooth isomorphism
We next move to the obstruction sheaves of various moduli spaces of interests. We
continue to denote by X a smooth complex threefold.
Let IZi be the the ideal sheaves of Z1 , · · · , Zk ⊂ U0[[α]] ×X U , which are part of the
universal family of X [[α]] ; let π1 be the first projection of X [[α]] × X . Then the obstruction
sheaf of X [[α]] is
(18)
Ob[[α]]
X
=
k
M
Ext 2π1
IZi , IZi
0
∼
=
i=1
Similarly, the obstruction sheaf of
(19)
Ob[[α]]
V0 =
k
M
k
M
π1∗ Ext 2 OZi , IZi
i=1
V0[[α]]
is defined to be the relative extension sheaf
Ext 2π1 IWi , IWi
0
i=1
∼
=
k
M
π1∗ Ext 2 OWi , IWi
i=1
in which IWi are ideal sheaves of the subschemes W1 · · · , Wk ⊂ V0[[α]] ×X V that are
part of the universal family of V0[[α]] ; that π1 is the first projection of V0[[α]] ×X V .
These obstruction sheaves are related in the obvious way. First, we let
A1 = AX [[α]] ×X/X [[α]]
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and
A2 = AV [[α]] ×X V/V [[α]]
0
0
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Zero dimensional Donaldson–Thomas invariants of threefolds
respectively be the sheaf of smooth functions on X [[α]] × X and V0[[α]] ×X V that
are analytic along fibers of X [[α]] × X/X [[α]] and V0[[α]] ×X V/V0[[α]] . Because Ψα of
Lemma 2.6 is induced by the universal family and because the tautological map
h = (Ψα , p2 ◦ ψ) : V0[[α]] ×X V −→ X [[α]] × X,
where p2 ◦ ψ : V → X is the composite of ψ : V → X × X with the second projection
of X × X , maps fibers to fibers and is analytic along fibers,
(20)
h∗ IZi ⊗ A1 ∼
= IWi ⊗ A2 .
Here the tensor products are over the sheaves OX [[α]] ×X and OV [[α]] ×X V . On the other
0
hand, by the property of relative extension sheaf, the tensor product
Ob[[α]]
V0 ⊗ AV [[α]]
0
∼
=
k
M
π1∗ Ext 3A2 (IWi , IWi ) ⊗ A2
i=1
∼
=
k
M
π1∗ Ext 3A2 IWi ⊗ A2 , IWi ⊗ A2 .
i=1
Here the extension sheaf Ext 3A2 (·, ·) is defined using locally free resolution of locally
free sheaves of A2 –modules.
Because of the isomorphism (20), the last term in the above isomorphisms is canonically
isomorphic to
∼
=
k
M
π1∗ Ext 3A2 h∗ (IZi ⊗ A1 ), h∗ (IZi ⊗ A1 )
i=1
∼
=
k
M
[[α]]
π1∗ Ext 3A1 OZi ⊗ A1 , IZi ⊗ A1 ∼
= Ψ∗α ObX ⊗ AX [[α]] .
i=1
This proves that
(21)
[[α]]
Ψ∗α ObX[[α]] ⊗ AX [[α]] ∼
= ObV0 ⊗ AV [[α]] .
0
4
Representing Virtual Cycles
We will begin this section with a quick review of the construction of the virtual cycle of
a scheme with perfect obstruction theory. For more details of this construction, please
consult Behrend and Fantechi [4] or Li and Tian [7].
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4.1
Jun Li
Virtual cycles via Gysin map
For the moment, we assume W is a quasi-project scheme with a perfect obstruction
theory and obstruction sheaf Ob. Following [4, 7], the virtual cycle of W is constructed
via
(1) finding a vector bundle V on W and a surjective sheaf homomorphism3 V → Ob;
then the obstruction theory of W provides us a unique cone cycle4 C ⊂ V of
codimension rkV ;
(2) defining the virtual cycle [W]vir = 0∗V [C] via the Gysin homomorphism
0∗V : H∗BM (V, Z) → H∗ (W, Z).
Since every subvariety of V defines a class in the Borel–Moore homology group,
0∗V [C] is well-defined.
Before we move on, a few comments are in order.
Usually, the Gysin map is defined as a homomorphism between Chow groups. For
us, we shall use the Borel–Moore homology group and use intersecting with smooth
sections to define this homomorphism.
The cone cycle C ⊂ V is unique in the following sense. To each closed w ∈ W , we let
ŵ be the formal completion of W along w and fix an embedding
ˆ Spec k[[T W ∨ ]]
ŵ ⊂ T ,
w
that is consistent with the tangent space at their only closed points. We let O =
Ob ⊗OW k(w) be the obstruction space to deforming w in W . Then the Kuranishi map
of the obstruction theory provides a canonical embedding of the normal cone Cŵ T̂ to ŵ
in T̂ :
Cŵ T̂ ⊂ ŵ × O,
where the later is viewed as a vector bundle over ŵ. The uniqueness of C asserts that
there is a vector bundle homomorphism
ηw : V ×W ŵ −−→ O × ŵ
extending the homomorphism V|w → O induced by V → Ob such that
(22)
3
η ∗ Cŵ T̂ = C ∩ (V ×W ŵ),
In this paper, whenever we use a Roman alphabet, say V , to denote a vector bundle, we will
use its counter part V to denote
P the sheaf of regular sections OW (V).
4
A cycle is a finite union
mi Di of subvarieties Di with integer coefficients mi ; it is a cone
cycle if all Di are cones in V .
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Lastly, the resulting cycle 0∗V [C] ∈ A∗ W is independent of the choice of V → Ob.
As we will see in the later part of this paper, it will be useful to eliminate the dependence
of constructing [W]vir on the choice of V → Ob. To achieve this, we will use smooth
sections of Ob to define the cycle [W]vir .
Before we do that, we shall first recall the notion of pseudo-cycles.
4.2
Pseudo-Cycles
In this work, we shall use pseudo-cycles to represent homology classes in a stratifiable
space.
Let Θ ⊂ W be a triangulable closed subset of a stratified space W . We shall fix a
Riemannian metric5 on W and denote by Θ the –tubular neighborhood Θ in W .
Definition 4.1 A (Θ, )–relative d –dimensional pseudo-chain is a pair (f , Σ) of a
smooth, oriented d –dimensional manifold with smooth boundary ∂Σ and a continuous
map f : Σ → W such that f is smooth over Σ − f −1 (Θ ).
We denote the Z–linear span of all such pseudo-chains by PChd (W)Θ with –implicitly
understood.
We define an equivalence relation ∼
= on PChd (W)Θ as follows. For notation brevity,
we shall use [f ]pc to denote the pseudo-cycle (f , Σ) with Σ implicitly understood. We
call two such chains [f1 ]pc and [f1 ]pc equivalent if there are open subsets Ai ⊂ Σi and
orientation preserving diffeomorphism φ : A1 → A2 so that Σi − Ai ⊂ fi−1 (Θ ) for
i = 1 and 2, and f2 ◦ φ|A1 = f1 |A1 ; we define [f1 ∪ f2 ]pc ∼
= [f1 ]pc + [f2 ]pc with the union
[f1 ∪ f2 ]pc be the map Σ1 ∪ Σ2 → W induced by f1 and f2 ; we define [−f ]pc ∼
= −[f ]pc
with [−f ]pc be f : Σ− → W and Σ− the space Σ with the opposite orientation. The
equivalence relation ∼
= generates an ideal in the Abelian group PChd (W)Θ .
There is an obvious boundary homomorphism
∂ : PChd (W)Θ −−→ PChd−1 (W)Θ
that sends any [f ]pc to [∂f ]pc with ∂f is the restriction of f to ∂Σ. The kernel of its
induced homomorphism is defined to be the space of pseudo-cycles relative to Θ:
PCyd (E)Θ , ker{∂ : PChd (W)Θ −→ PChd−1 (W)Θ / ∼
=}.
The proof of the following lemma is standard.
5
We can embed W in a smooth space and use the induced Riemannian metric on the ambient
space.
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Jun Li
Lemma 4.2 Suppose dimR Θ ≤ d − 2 and suppose the –tubular neighborhood Θ is
a deformation retract to Θ. Then every (Θ, )–relative pseudo-cycle [f ]pc ∈ Pcyd (W)Θ
defines canonically a homology class [f ] ∈ Hd (W, Z).
One version of pseudo-cycle we shall repeatedly use is the following:
Remark 4.3 Any pair (B, Θ) of a closed subset B ⊂ W and a stratifiable closed Θ ⊂ W
such that B − Θ is a smooth, oriented d –dimensional manifold and dimR Θ ≤ d − 2 is
a (Θ, )–relative pseudo-cycle for all > 0.
This can be seen as follows. For any > 0, we pick an open O ⊂ B ∩ Θ/2 so
that Σ = B − O is a smooth manifold with smooth boundary. Then the identity map
f : Σ → B ⊂ W is a (Θ, )–relative pseudo-cycle.
In case W is stratifiable, which is the case when W is an open subset of a quasiprojective scheme, Θ deformation retract to Θ for all sufficiently small . Because
dimR Θ ≤ d − 2, for all sufficiently small these (Θ, )–relative pseudo cycles all
represent the same homology class in Hd (W, Z). Because of this, in the future we will
call B a Θ–relative pseudo-cycle and will denote by [B] ∈ Hd (W, Z) the resulting
homology class. In case Θ is the singular locus of B, we shall call B a pseudo-cycle
directly with Θ = Bsing implicitly understood.
4.3
Cycle representatives via smooth sections of sheaves
We now investigate how to intersect with smooth sections to define the Gysin homomorphism of a cone cycle in a vector bundle over W .
Let C be a pure C–dimension r algebraic cone cycle in a vector bundle π : V → W ;
X
C=
mi Ci
its irreducible components decomposition. We pick a stratification S of W and a
stratification S 0 of ∪i Ci so that the induced map ∪i Ci → W is a stratified map. Also,
to each stratum S0 ∈ S 0 with S = π(S0 ) its image stratum, we shall denote by VS the
restriction to S of V ; for a smooth section s ∈ AS (V), we denote its graph by Γs ⊂ V .
Definition 4.4 A smooth section s ∈ AS (V) is said to intersect transversally with C if
every stratum S0 ⊂ ∪i Ci , which is a smooth subset in VS , intersects transversally with
Γs ∩ VS inside VS .
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By embedding W in a projective space and extending V , we can apply the standard
Sard’s transversality theorem to conclude that the set of smooth sections that intersect
transversally with C is dense in the space of all smooth sections.
Let s be a section that intersects transversally with C. We claim that the intersection
Γs ∩ C is a pseudo-cycle. Indeed, let Si ⊂ Ci be the open stratum of Ci ; then Si ⊂ Ci
is smooth, open and Zariski dense; therefore, dimR Ci − Si ≤ dimR Ci − 2. Suppose
dim Ci = d (for all i) and rkV = r. Then Γs ∩ Si is a smooth, oriented manifold of
real dimension 2d − 2r; its complement Γs ∩ (Ci − Si ) has real dimension at most
2d − 2r − 2. According to Remark 4.3, by taking
Θ = ∪i Γs ∩ (Ci − Si ),
the set Γs ∩ Ci becomes a 2d − 2r–dimensional pseudo-cycle (relative to Θ). We
denote this pseudo-cycle by (Γs ∩ Ci )pc and denote its image pseudo-cycle under the
projection π : V → W by π∗ (Γs ∩ Ci )pc . Finally, by linearity,
X
X
(Γs ∩ C)pc =
(Γs ∩ Ci )pc and π∗ (Γs ∩ C)pc =
π∗ (Γs ∩ Ci )pc .
i
i
It is immediate to check that its associated homology class given by Lemma 4.2 is the
Gysin homomorphism image of [C]
0∗V [C] = [π∗ (Γs ∩ C)pc ] ∈ H2d−2r (W, Z).
We now apply this technique to construct the virtual cycle [W]vir , assuming that W is
quasi-projective with a perfect obstruction theory and the obstruction sheaf Ob. As we
mentioned, we first pick a locally free sheaf V and a quotient sheaf homomorphism
V → Ob, thus obtaining a cone C ⊂ V given by the obstruction theory of W . We then
pick the standard pair of stratification S 0 of C and S of W that respects the morphism
C → W and the loci of non-locally freeness of Ob. Then as was argued, we can find a
section s ∈ AS (V) that intersects transversally with C. The virtual fundamental cycle
[W]vir , which is 0∗V [C], is the homology class given by the pseudo-cycle
[W]vir = 0∗V [C] = [π∗ (C ∩ Γs )pc ] ∈ H∗ (W, Z).
We next show that such pseudo-cycle can be constructed using the image section
ξ ∈ AS (Ob) of s under the homomorphism AS (V) → AS (Ob). To this end, we need
first to recover the pseudo-cycle π∗ (Γs ∩ C)pc using the smooth section ξ ∈ AS (Ob).
Suppose V 0 → Ob is a surjective homomorphism and t ∈ AS (V 0 ) is a lift of ξ under
AS (V 0 ) → AS (Ob), which exists by the exact sequence at the end of subsection 2.2.
We claim that t intersects transversally with the virtual normal cone C0 ⊂ V 0 and
(23)
π∗ (Γs ∩ C)pc = π∗0 (Γt ∩ C0 )pc
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Jun Li
as pseudo-cycles in W .
We first prove the case where V 0 = V and t ∈ AS (V) is another lifting of ξ . We
first show that as sets π(Γs ∩ C) = π 0 (Γt ∩ C0 ). For this, we consider the section
s − t ∈ AS (V) and its induced fiberwise translation
`s−t : V → V;
x ∈ Vw 7−→ x + s(w) − t(w) ∈ Vw .
Because of [7], the fiber C ∩ Vw of the cone over w is translation invariant under vectors
in
Kw = ker{Vw → Ob|w }.
Therefore `s−t : V → V maps C to C. But on the other hand, `s−t maps Γt to Γs ,
hence `s−t (Γt ∩ C) = Γs ∩ C, which proves (23).
It remains to show that t intersects C transversally and the induced orientation on
π(Γs ∩ C) coincides with that of π(Γt ∩ C). First, because the stratification S respects
the non-locally freeness of Ob, the restriction Ob ⊗OW OS is locally free. Hence the
collection {Kw | w ∈ S} forms a subbundle of VS . By the translation invariance of
C ∩ Vw under Kw and by the minimality of the stratification S 0 , S0 ∩ Vw is invariant
under the translations by vectors in Kw ; hence S0 = `s−t (S0 ).
On the other hand, since the map `s−t : VS → VS is a smooth diffeomorphism, Γt
intersects transversally with S0 if and only if its image `s−t (Γt ) = Γs intersects
transversally with `s−t (S0 ) = S0 , which is true by our choice of s. Hence Γt intersects
transversally with S0 . In particular, `s−t (Γt ∩ C) = Γs ∩ C and hence π∗ (Γt ∩ C)pc =
π∗ (Γs ∩ C)pc .
Next we consider the general situation where V 0 → Ob is an arbitrary quotient sheaf
homomorphism by a locally free sheaf. We claim that by picking a lifting t ∈ AS (V 0 )
of ξ ∈ AS (Ob), we obtain the identical cycle representatives of [W]vir . Indeed, by our
previous discussion, we only need to consider the case that V is a quotient sheaf of V 0
and V 0 → Ob is the composite of
V 0 −−→ V −−→ Ob.
In this case, the cone cycle C0 ⊂ V 0 is merely the pull back of C ⊂ V via the induced
vector bundle homomorphism ϕ : V 0 → V . Now let t0 ∈ AS (V 0 ) be a lifting of
s ∈ AS (V), and let π 0 : V 0 → W be the projection. Then because C0 = ϕ∗ C,
π∗0 Γt ∩ ϕ∗ C pc = π∗0 Γt0 ∩ ϕ∗ C pc = π∗ Γs ∩ C pc
as pseudo cycles. This shows that the pseudo-cycle representative π Γs ∩ C of [W]vir
only depends on the section ξ ∈ AS (Ob).
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This way, those sections ξ ∈ AS (Ob) whose lifts intersect transversally with the
normal cone provide us pseudo-cycle representatives of the virtual cycle [W]vir . In the
following, we shall denote such representative by D(ξ)pc .
In the remainder of this paper, for a scheme W with obstruction sheaf Ob, we shall fix
a locally free sheaf OW (V) that surjects onto Ob with C the virtual normal cone in V
of the obstruction theory of W . We say that a smooth section ξ ∈ AS (Ob) is a good
section if it has a lift s ∈ AS (V) that intersects transversally with C. By the previous
construction, the image π∗ (Γs ∩ C)pc defines a closed pseudo-cycle D(ξ)pc .
We summarize this subsection in the following Proposition.
Proposition 4.5 Let the notation be as before. Then any good section ξ ∈ AS (Ob)
defines a pseudo-cycle D(ξ) in W whose associated homology class is the virtual cycle
[W]vir in H∗ (W, Z).
4.4
Virtual cycle of Hilbert schemes of α–points
We shall employ smooth sections to construct cycle representatives of the virtual cycles
of Y [[α]] for a smooth family of quasi-projective threefolds Y/T over a smooth base T .
But before we do that, we shall first define how cycles are pulled back by the tautological
map Y [[α]] → Y [α] .
We begin with defining the α–multiplicity of the symmetrization morphism
Sα : Y Λ −→ Y (α) .
Let x ∈ Y Λ be any element. We define the multiplicity mα (x) be the number of
permutations of Λ that fix x and leave α invariant. Namely,
mα (x) = #{σ ∈ Symm(Λ) | σ(α) = α, σ(x) = x}.
It is easy to see that all elements in (Sα )−1 Sα (x) have identical α–multiplicities; their
summations satisfies6
X
(24)
mα (y) = α!.
y∈(S α )−1 S α (x)
For any z ∈ Y [[α]] lies over x ∈ Y Λ , we define its α–multiplicity mα (z) = mα (x).
6
We define α! = α1 ! · · · αk !.
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Because the virtual normal cone is constructed based on the obstruction theory of the
Hilbert scheme of points IY/T (0, n), we need to work with the tautological morphism
ϕα : Y [[α]] −−→ IY/T (0, α)
with
IY/T (0, α) = IY/T (0, m1 ) ×T · · · ×T IY/T (0, mk ),
where mi = |αi |.
We let F be a vector bundle on IY/T (0, α) and let C ⊂ F be a cycle, which is a linear
combination of subvarieties of F . We let E be the pull back vector bundle ϕ∗α F on
Y [[α]] ; let φα : E → F and πα : E → Y [[α]] be the obvious projections.
Definition 4.6 For any subvariety D ⊂ E we define the α–multiplicity mα (D) of D be
the α–multiplicity of the general point of the image πα (D) ⊂ Y [[α]] . For any subvariety
r
C ⊂ F and irreducible decomposition φ−1
α (C) = ∪i=1 Di , we define the pull-back
φ∗α C
=
r
X
mα (Di )Di .
i=1
We define the pull back of any cycle by extension via linearity.
The identity (24) implies that for any cycle C in F
φα∗ φ∗α C = α! C.
The virtual cycle of Y [[α]] will be defined as the Gysin map image of the pull back virtual
normal cone given by the obstruction theory of IY/T (0, α). Let Fα be a locally free
sheaf on IY/T (0, α) that makes the obstruction sheaf of IY/T (0, α) its quotient sheaf.
Then the obstruction theory of IY/T (0, α) provides us a cone cycle Cα ∈ A∗ Fα in the
vector bundle Fα associated to Eα . We let Eα be the pull back vector bundle over
Y [[α]] ; let φα : Eα → Fα be the projection, and let
C α = φ∗ C α
be the pull back cycle in Eα defined in Definition 4.6. Since the pull back of the
obstruction sheaf of IY/T (0, α) is canonically isomorphic to the obstruction sheaf of
Y [[α]] , the sheaf Eα = O(Eα ) has the obstruction sheaf Ob[[α]]
of Y [[α]] as its quotient
Y
sheaf. By abuse of notation, we will call the cycle Cα ∈ A∗ Eα the virtual cone of the
obstruction theory of Y [[α]] .
Definition 4.7 We define the virtual fundamental class
[Y [[α]] ]vir = 0∗Eα Cα ∈ H∗ (Y [[α]] , Z).
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Since the push-forward of [Y [[α]] ]vir under Y [[α]] → IY/T (0, α) is α! times the virtual
fundamental cycle [IY/T (0, α)]vir , it is independent of the choice of Eα , thus is welldefined.
To get an explicit cycle representative, we can take a good section ξα of Eα that
intersects the cone Cα transversally to form a pseudo-cycle
D(ξα ) = πα (Γξα ∩ Cα )pc ∈ PCy∗ (Y [[α]] ).
As was shown before, the cycle D(ξα ) only depend on the image section sα ∈ AS (Ob[[α]]
Y )
of ξα . Hence to eliminate the dependence on Eα , we shall denote D(ξα ) by D(sα ). We
have
[D(sα )] = [Y [[α]] ]vir ∈ H∗ (Y [[α]] , Z).
5
Approximation of Virtual cycles
[[β]]
[[α]] ∼
Because the partial equivalence Y(α,β)
= Y(β,α) is functorial, we expect that the
obstruction sheaves, the virtual normal cones, and the cycle representatives of the virtual
[[α]] ∼ [[β]]
cycles of Y [[α]] and Y [[β]] are identical over Y(α,β)
= Y(β,α) . It is the purpose of this
section to show that this is the case.
5.1
Cones under equivalence
Our immediate task is to compare the sheaves Ob[[α]]
with Ob[[β]]
and compare their
Y
Y
respective virtual normal cones. Since for different α and β , the mentioned partial
[[α∧β]]
[[β]]
[[α∧β]]
[[α]]
∼
equivalence follows from the equivalence Y(α,α∧β)
= Y(α∧β,α) and Y(β,α∧β) ∼
= Y(α∧β,β) ,
for our purpose we only need to investigate the case where α > β .
We first set up the notation. Let α > β be any pair. In this section we will fix once and
for all an indexing
α = (α1 , · · · , αk )
and
β = (β11 , · · · , β1l1 , · · · , βk1 , · · · βklk )
so that βi1 ∪ · · · ∪ βili = αi . Since α > β , such indexing exists.
[[α]]
We let Z̃1 , · · · , Z̃k be subschemes of Y(α,β)
×T Y that are part of the universal family
[[β]]
[[α]]
of Y [[α]] over Y(α,β)
; we let W̃11 , · · · , W̃klk be subschemes of Y(β,α)
×T Y that are part
[[β]]
of the universal family of Y [[β]] over Y(β,α)
.
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[[α]] ∼ [[β]]
The partial equivalence Y(α,β)
= Y(β,α) induces a rational map from IY/T (0, β) to
IY/T (0, α). Let
[[α]]
%α : Y(α,β)
−−→ IY/T (0, α)
be the tautological morphism that is induced by the families (Z1 , · · · , Zk ); we let
β
U(α,β)
⊂ IY/T (0, α) be the image subset of this map, which is open. Because of this, we
shall endow it with the induced scheme structure (usually non-reduced) from that of
IY/T (0, α). For β , we have the similarly defined
[[β]]
%β : Y(β,α)
−−→ IY/T (0, β)
β
induced by the families Wij . We then endow the open subset U(β,α)
= Im(%β ) with the
induced scheme structure from that of IY/T (0, β).
β
[[β]]
More to that, for any η̃ ∈ U(β,α)
over t ∈ T that is the image of an η ∈ Y(β,α)
, the
associated (indexed) zero-subschemes η11 , · · · , η1l1 , · · · , ηklk ⊂ Yt for each 1 ≤ i ≤ k
have that the collection ηi1 , · · · ηili is mutually disjoint. Hence we can assign
ξi = ηi1 ∪ · · · ∪ ηili ,
thus obtaining a zero-subscheme in IY/T (0, |αi |) and the tuple
(ξ1 , · · · , ξk ) ∈ IY/T (0, α).
β
It is easy to see that this correspondence defines a one–one onto map from U(β,α)
to
α
U(α,β)
.
The proof given in subSection 1.3 immediately shows that
Lemma 5.1 The induced map
β
α
Υαβ : U(β,α)
−−→ U(α,β)
is an étale morphism between two schemes; it commutes with the maps %α , the map %β
[[α]] ∼ [[β]]
and the partial equivalence Y(α,β)
= Y(β,α) . Further, the obstruction sheaves Ob[[α]] of
β
IY/T (0, α) are isomorphic under pull back by U(β,α)
:
[[α]] α
∼
Υ∗αβ Ob[[β]]
.
= ObY |Y(α,β)
Y
The virtual normal cones are also identical under this isomorphism. We pick a locally
α
α
free sheaf Eα on U(α,β)
that makes Ob[α]
Y its quotient sheaf (over U(α,β) ). Then
β
Eβ = Υ∗αβ Eα is a locally free sheaf over U(β,α)
that makes Ob[β]
Y its quotient sheaf.
Then the perfect-obstruction theory provides us the virtual normal cone Cα ⊂ Eα and
the normal cone Cβ ⊂ Eβ .
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Lemma 5.2 Under the induced flat morphism φαβ : Eβ → Eα , the cycles
φ∗αβ Cα = Cβ .
Proof This follows from the uniqueness assertion on cones (22) and the following
invariance result.
Lemma 5.3 Let U ⊂ X be an open subset and let ξ ⊂ U be a zero subscheme.
Then the obstruction spaces to deforming ξ in U and in X are canonically isomorphic.
Further, under this isomorphism of obstruction spaces, the obstructions to deforming ξ
in U and in X are identical.
Proof First the to obstruction spaces are traceless extension groups
Ext2X (Iξ , Iξ )0
and
Ext2U (Iξ , Iξ )0 .
They are isomorphic because
Ext2X (Iξ , Iξ )0 ∼
= H 0 Ext 3OX (Oξ , Iξ ) = H 0 Ext 3OU (Oξ , Iξ ) ∼
= Ext2U (Iξ , Iξ )0 .
As to the obstruction theory, say using locally free resolutions of Iξ and using Cěch
cohomology representative of the obstruction classes, one checks directly that the
obstruction to deforming ξ in X gets mapped to the obstruction class to deforming ξ in
U under the canonical homomorphism
Ext2X (Iξ , Iξ )0 −−→ Ext2U (Iξ , Iξ )0 .
But because this arrow is an isomorphism, the two obstruction classes must be
identical.
5.2
Some further notations
From now on, for any α we fix a locally free sheaf Eα over IY/T (0, α) that makes
its quotient sheaf the obstruction sheaf Ob[α] of IY/T (0, α); we let Cα ⊂ Eα be
the associated virtual normal cone. Because Y [α] is IY/T (0, α) with reduced scheme
structure, we can view Eα as a vector bundle over Y [α] and view Cα as a cone cycle in
A∗ Eα .
For the Hilbert scheme of α–points Y [[α]] , its obstruction sheaf Ob[[α]]
is canonically
Y
∗
[α]
isomorphic to the pull back sheaf ρα Ob ⊂ IY/T (0, α) under the tautological morphism
ρα : Y [[α]] → Y [α] .
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Thus by taking Ẽα = ρ∗α Eα , the obstruction sheaf Ob[[α]]
naturally becomes a quotient
Y
sheaf of Ẽα .
We let Ẽα → Eα be the tautological projection, viewed as a stack flat morphism
Ẽα = Eα ×Y (α) Y α −−→ EΛ .
The normal cone C̃α ⊂ Ẽα is then defined to be the stack flat pull back of Cα as
specified in Definition 4.6.
For stratifications of C̃α and Y [[α]] , we shall take the standard pair of stratifications that
respects the morphism C̃α → Y [[α]] and the loci of non-locally freeness of the sheaf
Ob[[α]]
Y .
Definition 5.4 We say that a smooth section sα ⊂ A(Ob[[α]]
Y ) intersects transversally
with its normal cone if one (thus all) of its lifts ξα ∈ A(Ẽα ) intersects transversally with
the cone C̃α .
Following the discussion in subSection 4.3, we can find sections of A(Ob[[α]]
Y ) that
intersect the normal cone transversally. For such sα , we shall denote by D(sα ) ⊂ Y [[α]]
the pseudo-cycle that is the image in Y [[α]] under Ẽα → Y [[α]] of intersecting a lift of sα
with C̃α ⊂ Ẽα .
Now let sα ∈ A(Ob[[α]]
Y ) be a section that intersects transversally with the normal cone.
[[β]]
∼
Because Ob[[α]]
|
Y
Y [[α]] = ObY |Y [[β]] canonically, we can view sα|β = sα |Y [[α]] as a
(α,β)
section of
of
Ob[[β]]
Y
5.3
Ob[[β]]
Y
over
(β,α)
[[β]]
Y(β,α)
.
(α,β)
Because of Lemma 5.1 and 5.2, sα|β is a smooth section
and intersects transversally with the normal cone of Y [[β]] .
Compatible cycle representatives
To compare the cycles [Y [[α]] ]vir , in this section we shall carefully pick smooth sections
sα so that for any pair α and β the cycle representatives D(sα ) are D(sβ ) are identical
[[α]] ∼ [[β]]
over most part of the intersection Y(α,β)
= Y(β,α) .
To this end, we form the strict α–diagonal
∆α = {x ∈ Y Λ | a ∼α b ⇒ xa = xb };
they are closed; for different α and β , ∆α ∩ ∆β = ∆α∨β 7 . Next we fix a sufficiently
small c > 0 and pick a function ε : PΛ → (0, c) whose values on any ordered pair
7
α ∨ β is the smallest element among all that are larger than or equal to both α and β .
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Zero dimensional Donaldson–Thomas invariants of threefolds
α > β obey ε(α) > Rε(β) for a sufficiently large R. After fixing a Riemannian metric
on Y , we then form the ε–neighborhoods of ∆α ⊂ Y Λ :
∆α,ε = {x ∈ Y Λ | dist(x, ∆α ) < ε(α)}.
For any β ≤ α, we form
[
∆αβ,ε =
∆γ,ε
and
[
Qαβ,ε = ∆β,ε −
α≥γ≥β
∆αγ,ε .
α≥γ>β
Note that Qαβ,ε are closed subsets of ∆β,ε .
We have the following intersection property of these sets.
Lemma 5.5 For any pair β1 , β2 ≤ α satisfying ∆β1 ,ε ∩ Qαβ2 ,ε 6= ∅, necessarily
β2 ≥ β1 .
Proof Because c is sufficiently small, whenever ∆µ,ε ∩ ∆ν,ε 6= ∅, necessarily
∆µ ∩ ∆ν 6= ∅. Then because ∆µ ∩ ∆ν = ∆µ∨ν , because ∆µ intersects ∆ν
perpendicularly, and because ε(µ ∨ ν) > R2 ε(µ) + R2 ε(ν),
∆µ,ε ∩ ∆ν,ε ⊂ ∆µ∨ν,ε .
Now suppose β2 β1 , then β1 ∨ β2 > β2 ; therefore we have ∆β1 ,ε ∩ ∆β2 ,ε ⊂ ∆β1 ∨β2 ,ε
and Qαβ2 ,ε ⊂ ∆αβ2 ,ε − ∆αβ1 ∨β2 ,ε . Combined, we have
∅=
6 ∆β1 ,ε ∩ Qαβ2 ,ε ⊂ ∆β1 ∨β2 ,ε ∩ ∆β2 ,ε − ∆β1 ∨β2 ,ε = ∅.
This proves β2 ≥ β1 .
`
An immediate corollary of this is that ∆αβ,ε = α≥γ≥β Qαγ,ε forms a partition (a disjoint
union) of ∆αβ,ε . By choosing β = 0Λ , it also shows that {Qαβ,ε | β ≤ α} forms a
partition of Y Λ .
Moving to Y [[α]] , we form
α
α
Nβ,ε
= ρ−1
α Nβ,ε
α
and Qαβ,ε , ρ−1
α Qβ,ε
⊂ Y [[α]] ,
in which we continue to denote by ρα : Y [[α]] −→ Y Λ the projection. The collection
{Qαβ,ε | β ≤ α} forms a partition of Y [[α]] .
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Lemma 5.6 For sufficiently small c, we can find a collection of sections sα ∈
AS (Ob[[α]] ) that satisfy the properties
(i) each sα intersects transversally with the normal cone of Ob[[α]]
Y ;
(ii) for any β < α, the sections sα and sβ coincide over Qαβ,ε .
The requirement (ii) is understood as follows. To each β < α, because Qαβ,ε is disjoint
[[α]]
α
from ρ−1
α (∆γ ) for all α ≥ γ > β , it lies inside Y(α,β) . Thus restricting to Qβ,ε both
sα and sβ are sections of the same sheaf, and hence can be said to equal.
Proof We prove the lemma by induction. Because the space Y [[α]] is a disjoint union
of Qαβ,ε , we will construct sα by specifying its values along each of the above subsets
according to (ii) and then showing that the resulting section can be extended to satisfy
(i).
We now construct the section sα by induction. Suppose we have already constructed sβ
`
for all β < α that satisfy the properties (i)–(ii). Along the partition Y [[α]] = β≤α Qαβ,ε ,
we shall follow the rule (ii) to define
sα|β = sβ |Qα(β,α) ∈ Γ(Qαβ,ε , AS (Ob[[α]]
Y )
be the restriction to Qαβ,ε of sβ . Inductively, this will define
α
sα ∈ Γ Y [[α]] − Nα,ε
, AS (Ob[[α]]
Y )
α .
after checking that the collection sα|β forms a smooth section over Y [[α]] − Nα,ε
α is a disjoint union of {Qα |
We now prove that it is so. First, because Y [[α]] − Nα,ε
β,ε
α must lie in a Qα for a unique β < α. In case z is
β < α}, each z ∈ Y [[α]] − Nα,ε
β,ε
an interior point of Qαβ,ε , then sα coincide with sβ near z; by induction hypothesis,
sα satisfies the requirement (i) and (ii) near z. In case z is not an interior point of
α is open, by Lemma 5.5 this is possible only when z lies in the closure
Qαβ,ε , since Nγ,ε
cl Qαβ0 ,ε for some β 0 > β . There are two possibilities: one is when β 0 = α, in which
case nothing to prove. The other is when β 0 < α. Since sα is defined via sβ on Qαβ,ε
and via sβ 0 on Qαβ0 ,ε , to check the continuity of sα , we need to compare the germ of sβ
0
[[α]]
and of sβ 0 near z. In this case, Qαβ0 ,ε ⊂ Y(α,β)
, thus can be considered lies in Y [[β ]] ; as
0
0
to Qαβ,ε , Qαβ,ε ∩ Y [[β ]] = Qββ,ε . Hence by induction hypothesis, sβ 0 is an extension of
sβ , Thus sα is well-defined near z, and thus is smooth near z.
α , which is closed in Y [[α]] , we can
Once we know that sα is smooth over Y [[α]] − Nα,ε
[[α]]
extend it to a good smooth section of ObY over Y [[α]] , which completes the proof of
the Lemma.
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From now on, we fix such a collection {sα }α∈PΛ and form their associated pseudo-cycle
representatives D(sα ).
5.4
Approximation of the virtual cycles
In this subsection, we shall investigate how pseudo-cycles D(sα ) are related by looking
at their images in Y Λ .
We let d be the real dimension of T ; let D(sα ) be the d –dimensional pseudo-cycles
constructed relative to a set Θ̃α ⊂ Y [[α]] of dimension ≤ d − 2. We take Θ ⊂ Y Λ be the
union of the images under the projection ρα : Y [[α]] → Y Λ of all Θ̃α : Θ = ∪α ρα (Θ̃α ).
Then to a sufficiently small > 0 independent of c, we pick f̃α : Σα → Y [[α]] a
representative of D(sα ) as d –dimensional pseudo-cycle relative to (ρ−1
α (Θ), ). The
composition fα = ρα ◦ f̃α defines a (Θ, )–relative pseudo-cycle, which we denote by
[fα ]pc .
We now relate different pseudo-cycles [fα ]pc by forming inductively
X
(25)
δα = [fα ]pc −
δβ ∈ Cycd (Y Λ )Θ .
β<α
We have the following vanishing result:
Lemma 5.7 The pseudo-chain δα ∩ (Y Λ − ∆α,2c ) is equivalent to 0.
Proof What we need to show is that to any z ∈ Y Λ − (∆α,2c ∪ Θ ) we can find
a sufficiently small ball Br (z) centered at z so that δα ∩ Br (z) as a pseudo-cycle is
equivalent to zero.
We now prove this by induction. For α = 0Λ , there is nothing to prove. Now let
α ∈ PΛ be any element so that this holds true for all β < α. Let x ∈ Y Λ be any point
away from ∆α,2c ∪ Θ and let β < α be so that
(26)
x ∈ Qαβ,ε .
We first claim that for sufficiently small r, δγ ∩ Br (x) ∼ 0 for all β 6≥ γ < α.
Suppose not, then by induction hypothesis, x ∈ ∆γ,2c . Since c is sufficiently small
and ε(β1 ) > Rε(β2 ) for any β1 > β2 , we have ∆β,ε ∩ ∆γ,ε ⊂ ∆β∨γ,ε , and hence
ρ(z) ∈ ∆β∨γ,ε . Because we have assumed that β 6≥ γ , we must have β ∨ γ > β , which
implies that ∆β∨γ,ε ∩ Qαβ,ε = ∅, contradicting to x ∈ Qαβ,ε and in ∆β∧γ,ε .
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Because of this, when we intersects δα with Br (x), for sufficiently small r (25) tells us
that
X
δα ∩ Br (x) = [fα ]pc ∩ Br (x) −
δγ ∩ Br (x).
γ≤β
On the other hand by definition
δβ ∩ Br (x) = [fβ ]pc ∩ Br (x) −
X
δγ ∩ Br (x).
γ<β
Hence if we can show that [fα ]pc ∩ Br (x) = [fβ ]pc ∩ Br (x), the above two identities will
force δα ∩ Br (x) = 0, exactly what we intend to prove.
For this we argue as follows: let z ∈ D(sα ) ∩ ρ−1
α (Br (x)). Because of (26), x ∈ ∆β,ε
[[β]]
[[α]]
and x 6∈ ∪β<γ≤α ∆γ ; thus z must lie in Y(α,β) and thus also in Y(β,α)
. Hence (ii) of
lemma (5.6) implies that
−1
D(sα ) ∩ ρ−1
α (Br (x)) = D(sβ ) ∩ ρβ (Br (x)).
For the same reason, the above identity holds in case x ∈ D(sβ ). This proves the
lemma.
5.5
Truncated discrepancy cycles
Because δα is equivalent to zero away from ∆α,2c , we can truncate it by intersecting it
with the closed subset ∆α,e with a general 2c < e < 3c: δαe = δα ∩ ∆α,e . It can also
be defined inductively by
X
δβe ,
δαe ∼ [fα ]pc ∩ ∆α,e −
β<α
where the intersection [fα ]pc ∩ ∆α,e can be replaced by any pseudo-chain fα |Σeα with
Σeα = fα−1 (∆α,e )−O for an open O such that Σeα has smooth boundary and fα (O) ⊂ Θε .
By Sards theorem, for general e the set fα−1 (∆α,e ) has smooth boundary away from
fα−1 (Θε ). Thus we can choose e that works for all α.
Lemma 5.8 With such choice of e and pseudo-chain representative [fα ]pc ∩ ∆α,e , the
inductively defined pseudo chain δαe is a pseudo-cycle relative to (Θ, ).
Proof This follows directly from Lemma 5.7 and the definition of pseudo-cycle.
In case α has more than one equivalence class, the cycle δα is equivalent to the product
of δαi . More precise, by viewing each αi as a set, we can form the spaces Y αi and the
pseudo-cycles δβi for βi ∈ Pαi . We continue to denote by αi the top partition in Pαi .
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Lemma 5.9 We can choose representatives δαi and δα so that under the identity
Q
Qk
Y Λ = kT Y αi as pseudo-cycles δα ∼
= i=1 δαi .
Proof For each αi , we form the cycle representatives D(γ) ⊂ Y [[γ]] and the discrepancy
cycles δγ by picking a collection of sections {sγ | γ ∈ Pαi } provided by the
previous lemma. For each β ≤ α in PΛ , we write β as (β11 , · · · βklk ) and form
βi = (βi1 , · · · , βili ), each is a partition in Pαi . Then
Y [[β]] = Y [[β1 ]] × · · · × Y [[βk ]]
1 ]]
k ]]
and Ob[[β]]
= π1∗ Ob[[β
⊕ · · · ⊕ πk∗ Ob[[β
Y
Y
Y
with πi the i-th projection of Y [[α]] to Y [[αi ]] . Further, the sections sβ1 , · · · , sβk provides
us a section
sβ = π1∗ sβ1 ⊕ · · · ⊕ πk∗ sβk
with associated pseudo-cycle
D(sβ ) = D(sβ1 ) × · · · × D(sβk ).
(27)
We claim that using such decomposition, the discrepancy pseudo-cycles
δβ ∼
= δβ1 × · · · × δβk .
(28)
Indeed, when β = 0Λ , then the identity reduces to
Y
D(sβ ) =
D(s1{a} ),
a∈Λ
which is (27). Now suppose the identity holds for all γ < β . Then
k
Y
D(sβi ) =
k X
Y
i=1 βij ≤βi
i=1
=
X
δβij =
X
δγ1 × · · · × δγk
(γ1 ,··· ,γk )∈Pβ1 ×···×Pβk
δγ + δ1β1 × · · · × δ1βk .
γ<β
The desired identity (28) then follows from D(sβ ) =
(27). This proves the Lemma.
P
γ<β δγ
+ δβ and the identity
By choosing a general e as before, the truncated discrepancy cycle also satisfies the
product formula (28) with δ· replaced by its truncated version δ·e .
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Proof of the Main theorem
The proof of the theorem now is fairly straightforward. We first apply the previous
construction to Y = X and to the set Λ = [n] of integers from 1 to n to relate the
degree of the virtual cycle
deg[IX (0, n)]vir =
1
deg[X [[n]] ]vir .
n!
(Here we follow the convention X [[n]] = X [[Λ]] .)
The right hand side can be expressed as the degree of an explicit zero-cycle. For this
[[n]] that intersects
we pick a smooth section s[n] of the obstruction sheaf Ob[[n]]
X of X
transversally with the normal cone. Because the virtual dimension of IX (0, n) is zero,
the resulting pseudo-cycle D(s[n] ) in X [[n]] is a zero-cycle, satisfying
deg[X [[n]] ]vir = deg[D(s[n] )].
To proceed, we shall relate it to the degrees of the discrepancy cycles δα . For a
sufficiently small and c and for all α ∈ PΛ we choose sections sα of Ob[[α]]
X
according to Lemma 5.6. We let D(sα ) in X [[α]] be the associated zero-cycle derived
by intersecting the normal cone by sα . However, because each δα is a zero-cycle,
n ⊂ Xn .
Lemma 5.7 shows that it is entirely contained in the –tubular neighborhood of X∆
Further, their degrees
X
deg[D(s[n] )] =
deg δα .
α∈PΛ
Thus to prove that deg[IX (0, n)]vir is expressible in terms of a universal polynomial in
Chern numbers of X we only need to show that the same hold for all deg δα , which then
follows from that of deg δ[n] by the product formula in Lemma 5.9. This way, we are
reduced to show that deg δ[n] is expressible by a universal expression in Chern numbers
of X .
Our next step is to use the construction of the diffeomorphism of a neighborhood of
0TX ⊂ TX with a neighborhood of the diagonal ∆(X) ⊂ X × X to transfer the cycle δ[n]
to the tangent bundle TX .
We let U ⊂ X × X be the tubular neighborhood of ∆(X) ⊂ X × X , let V ⊂ TX be the
tubular neighborhood of 0TX ⊂ TX and let ψ : V → U be the smooth isomorphism
provided by Lemma 2.4. The map ψ induces a smooth isomorphism
(29)
Ψα : V0[[α]] −−→ U [[α]] ⊂ X [[α]]
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[[α]]
onto an open neighborhood of X∆
⊂ X [[α]] (see Lemma 2.6). Because U [[α]] is an
[[α]]
open neighborhood of X∆ , by choosing c sufficiently small, Lemma 5.7 tells us that
the cycle δ[n] lies entirely in U [[n]] . Thus the degrees of
δ[n],V = Ψ∗[n] (δ[n] )
is identical to that of δ[n] .
The cycle δ[n],V can also be constructed using the universal family over the Grassmannian
of quotients CN → C3 . Let Q → Gr be the total space of the universal quotient bundle
and let Q[[α]] (resp. Q[[α]]
0 ) be the relative Hilbert scheme of α–points (resp. centered
α–points) of Q/Gr. To each Q[[α]]
we form its obstruction sheaf Ob[[α]]
0
Q ; we pick a
[[α]]
locally free sheaf Eα making ObQ its quotient sheaf; we let Eα be the associated
vector bundle and let Dα ⊂ Eα be the associated normal cone.
We then pick a smooth section tα of Eα for all α so that the collection {tα }α∈PΛ
satisfies the conclusion of Lemma 2.4. The sections tα provide us the pseudo-cycle
D(tα ) in Q[[α]]
relative to a subset Θ̃α of dimension at most 2 dim Gr − 8. According
0
to (25), the images under the projections ηα : Q[[α]]
→ Qn0 of the pseudo-cycles D(tα )
0
n
form the discrepancy pseudo-cycle δ[n],Q ⊂ Q0 relative to (Θ, ), the set Θ which is
the union of all ηα (Θ̃α ) and for which is sufficiently small.
The cycle δ[n],Q defines a codimension six homology class in Gr. Indeed, by Lemma 5.7,
the pseudo-cycle is zero outside the 2c–tubular neighborhood in Qn0 of the top diagonal
Qn∆ ∩ Qn0 . Because the top diagonal Qn∆ consists of points (ξ, · · · , ξ), they are centered
only if ξ = 0. Thus the top diagonal is the zero section of Qn0 /Gr; thus is compact.
Therefore, δ[n],Q is a compact pseudo-cycle, thus defines a homology class
[δ[n],Q ∈ HdimR Gr−6 (Qn0 , Z) ∼
= HdimR Gr−6 (Gr, Z).
We shall relate the cycle δ[n],Q with δ[n] in X n by picking a smooth map g : X → Gr
so that as smooth vector bundle g∗ Q = TX . Without loss of generality, we can choose
N to be sufficiently large and choose g to be an embedding. Obviously, g induces a
smooth isomorphism (compare to (12))
(30)
gn : Qn0 ×Gr X ∼
= (TX)n0 .
We have the following compatibility result.
Lemma 6.1 We can choose g and sections sα so that Qn0 ×Gr X intersects transversally
with δ[n],Q and the intersection
g∗n (δ[n],Q ) , δ[n],Q ∩ Qn0 ×Gr X ,
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Jun Li
considered as a cycle in (TX)n0 via the isomorphism (30), is identical to the cycle δ[n],V
via the inclusion V0n ⊂ (TX)n0 .
Once the lemma is proved, then we can use the constructed sections sα to form the
cycle δ[n] in X n to conclude
deg δ[n] = deg δ[n],V = deg g∗n (δ[n],Q ) = deg [δ[n],Q ]P.D. ∩ g∗ ([X]) ,
where [δ[n],Q ]P.D. is the Poincare dual of the homology class [δ[n],Q ] ∈ H6 (Gr, Z).
Because H ∗ (Gr, Z) is generated by ui ∈ H 2i (Gr, Z),
[δ[n],Q ]P.D. = Pn (u1 , u2 , u3 ) ∈ H 6 (Gr, Z)
expressible in a polynomial in ui . On the other hand, using the embedding Gr(N, 3) ⊂
Gr(N + 1, 3) and the proof that will follow, we see immediately that this polynomial Pn
is independent of the choice of N ; thus is universal in n.
Finally, because g∗n (ui ) = ci (X),
[δ[n],Q ]P.D. ∩ g∗ ([X]) = pn (c1 (X), c2 (X), c3 (X)) ∩ [X]
is a universal expression in the Chern numbers of X . This proves the main theorem.
We shall divide the proof of Lemma 6.1 into two steps: one is to compare the intersection
g∗n (δ[n],Q ) with a similarly constructed cycle δ[n],TX on (TX)n0 ; the other is to compare
the later with the pull back of δ[n] using the map V0n → X n . Since the proofs of both are
similar, we shall provide the details of the first while indicating the necessary changes
required for the second.
We begin with a quick account of the moduli of A3 . We let V = A3 and let V0[[α]] → V0n
be the tautological map from the Hilbert scheme of centered α–points to V0n that
was constructed in section 1 with Y replaced by V . Because GL(3) acts on V , it
acts on V [[α]] and V n . Further these actions leave the projection V [[α]] → V n and the
averaging map V n → V invariant, thus the space V0[[α]] = V [[α]] ×V 0 is GL(3)–invariant.
Not only that, its obstruction sheaf Ob[[α]]
and its obstruction theory are all naturally
V
GL(3)–linearized.
Because GL(3) is reductive, we can find a GL(3)–linearized locally free sheaf Fα on
V0[[α]] that makes Ob[[α]]
its GL(3)–equivariant quotient sheaf
V
(31)
Fα −−→ Ob[[α]]
V .
We let Fα be its associated vector bundle and let Dα ⊂ Fα be the associated normal
cone. Then each Dα,i in the irreducible decomposition
X
Dα =
mα,i Dα,i
i
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is a GL(3)–invariant cone-like subvariety of Fα .
We next fix a standard stratification of ∪i Dα,i → V0[[α]] subordinating to the loci of
the non-locally freeness of the sheaf Ob[[α]]
V . To each Dα,i , we let Sα,i ⊂ Dα,i be its
[[α]]
open stratum and let Tα,i ⊂ V0 be the image of Sα,i , which is a stratum of V0[[α]] .
Because Dα,i → Fα → V0[[α]] are GL(3)–equivariant, every stratum, including Sα,i , are
GL(3)–invariant.
Our next step is to use (31) to build locally free sheaves on Q[[α]]
and on (TX)[[α]]
making
0
0
their respective obstruction sheaves their quotient. For Q/Gr, we first cover Gr by
open Ua ⊂ Gr with vector bundle isomorphisms
∼
=
fa : Q ×Gr Ua −−→ V × Ua .
(32)
Then using the induced isomorphism
[[α]]
Q0[[α]] ×Gr Ua ∼
= V0 × Ua
and pV the first projection of the product on the right hand side, we can form the induced
(33)
[[α]]
∼
Eα,a = p∗V Fα −−→ pV Ob[[α]]
V = ObQ |Q[[α]] ×Gr Ua .
0
Over Uab = Ua ∩ Ub , the isomorphisms
fba = fb ◦ fa−1 : V × Uab −−→ V × Uab
induce transition isomorphisms
α
fba
: Eα,a |Q[[α]] ×Gr Uab −−→ Eα,b |Q[[α]] ×Gr Uab .
0
0
α also satisfy the cocycle condition. Hence f α
Since fab satisfy the cocycle condition, fab
ab
glue to form a locally free sheaf Eα on Q[[α]]
.
Obviously,
the
quotient
homomorphisms
0
(33) glue to form a quotient homomorphism
[[α]]
Eα −→ ObQ
.
The normal cone of Ob[[α]]
Q takes a simple form in this setting. Let Eα be the associated
vector bundle of Eα . The cycles
X
Dα × Ua =
mα,i Dα,i × Ua ∈ C∗ (Fα × Ua ) = C∗ (Eα |Q[[α]] ×Gr Ua )
0
i
glue together to form a cycle that is the normal cone Cα ⊂ Eα :
X
Cα =
mα,i Cα,i
i
with Cα,i the glued subvariety from Dα,i × Ua .
Geometry & Topology 10 (2006)
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Jun Li
Similarly, for the vector bundle TX/X we can carry over the same procedure to form a
locally free sheaf Ẽα over (TX)[[α]]
making the obstruction sheaf Ob[[α]]
TX its quotient
0
sheaf; the normal cone in the associate vector bundle C̃α ⊂ Ẽα is also the similarly
induced cycle by Dα ⊂ Fα .
Now we look at the smooth map g : X → Gr, which we assume to be an embedding,
and the smooth isomorphism
g0 : TX ∼
= Q ×Gr X ⊂ Q.
(34)
We let
gα : (TX)0[[α]] −−→ Q[[α]]
0
be the induced smooth map; we let A1 and A2 be the sheaves of smooth functions of
(TX)[[α]]
and Q[[α]]
respectively. We claim that there are smooth isomorphisms as shown
0
0
below that make the diagram commutative
g∗α Eα ⊗ A2 −−−−→ g∗α Ob[[α]]
Q ⊗ A2




∼
∼
(35)
=y
=y
Ẽα ⊗ A1
−−−−→
Ob[[α]]
TX ⊗ A1 .
Indeed, for any open Ua ⊂ X with trivialization TX|Ua ∼
= V × Ua and open Ub ⊂ Gr
∼
with trivialization Q|Ub = V × Ub and satisfying g(Ua ) ⊂ Ub , the isomorphism g0 in
(34) defines a smooth map h : Ua → GL(V) that makes the diagram commutative
g0
Q ×Gr X −−−−→


∼
=y
TX


∼
=y
(h,1)
V × Ua −−−−→ V × Ua
The family of automorphisms h then induce smooth diffeomorphism
(hα ,1)
V0[[α]] × Ua −−−−→

∼
=
y
V0[[α]] × Ua

∼
=
y
∼
=
Q[[α]]
×Gr Ua −−−−→ (TX)0[[α]] ×X Ua
0
and isomorphisms of sheaves
hα : Eα |Q[[α]] ×Gr Ua ∼
= Fα ⊗ OUa −−→ Fα ⊗ OUa ∼
= Ẽα |(TX)[[α]] ×X Ua ,
0
0
where the first and the third isomorphisms are induced by the construction of Eα and
Ẽα in (33) while the middle one is induced by h.
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2165
The homomorphisms hα for a covering of X patch together to form the left smooth
isomorphism in (35); it makes that diagram commutative.
This way, by lifting tα to a smooth section in Eα , then pulling the lifted section back to
a smooth section of Ẽα , and finally pushing forward the new section to a section in the
obstruction sheaf Ob[[α]]
TX , we obtain a smooth section. Obviously, the resulting section
the original pull back t̃α ; therefore, t̃α is smooth. Likewise, because g : X → Gr is a
smooth embedding, because each stratum of Cα,i is submersive onto Gr, and because
as subsets
Eα |Q[[α]] ×Gr X ⊃ Dα,i ∩ Q0[[α]] ×Gr X = D̃α,i ⊂ Ẽα
0
under the isomorphism (35), we see immediately that a lift of t̃α intersects transversally
with the cone cycle C̃α if and only if Q[[α]]
×Gr X intersects transversally with the
0
pseudo-cycle D(tα ). But this is possible if we choose g general. Therefore, we can
choose g that makes all pull back sections t̃α satisfy the first conclusion of Lemma 5.6.
Since the collection tα satisfies the second conclusion of Lemma 5.6, so does the
collection t̃α .
Apply the same argument to the standard embedding
GN : Gr(N, 3) −−→ Gr(N + 1, 3)
we see that we can choose sections tα ’s so that
δ[n],Gr(N,3) = g∗N (δ[n],Gr(N+1,3) )
are stable under GN . Therefore, their homology classes
[δ[n],Gr(N,3) ] ∈ H6 (Gr(N, 3); Z)
is stable under inclusions.
It remains to use the sections t̃α to get smooth sections sα of Ob[[α]]
X . First, because
[[α]]
[[α]]
V0 is an open subset of (TX)0 , the section t̃α restricts to a section of the obstruction
sheaf Ob[[α]]
of V0[[α]] . Without much confusion, we shall denote the restriction section
V
by t̃α as well.
Our next step is to take the induced sections s̃α of Ob[[α]]
X |U [[α]] under the smooth
isomorphisms
[[α]]
Ψ∗α ObX[[α]] ∼
= ObV
covering the smooth isomorphism Φα : V0[[α]] → U [[α]] ⊂ X [[α]] in (29) and show that
they are smooth and satisfy the conclusion of Lemma 5.6.
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Jun Li
We begin with more notations for X [[α]] . We pick a locally free sheaf Ēα on X [[α]]
making Ob[[α]]
its quotient sheaf; we let C̄α ⊂ Ēα be their associated normal cones
X
with irreducible decomposition
X
C̄α =
m̄α,i C̄α,i .
Like before, we denote by S̄α,i ⊂ C̄α,i its open stratum and denote by T̄α,i ⊂ X [[α]] the
image of S̄α,i .
Differing from the case of Q/Gr, we shall work with the restriction of the obstruction
sheaf to T̄α,i . To this end, we let
W̄α,i = ObX[[α]] ⊗OX[[α]] OT̄α,i ;
let W̄α,i be the associated vector bundle and let
ξ¯α,i : Ēα |T̄α,i −−→ W̄α,i
be the surjective homomorphism induced by Ēα → ObX[[α]] .
We shall do the same for (TX)[[α]]
0 . We let Wα,i be the associated vector bundle of the
restriction sheaf
[[α]]
W̃α,i = ObTX
⊗O [[α]] OT̃α,i ,
(TX)
0
which is locally free over T̃α,i ; we let
ξ˜α,i : Ẽα |T̃α,i −−→ W̃α,i
be induced by Ẽα → Ob[[α]]
TX .
Our next step is to show that possibly after re-indexing the i’s we have
(36)
Ψα (T̄α,i ∩ U [[α]] ) = T̃α,i ∩ V0[[α]] ;
and that under the canonical isomorphism Ψ∗α W̄α,i ∼
= W̃α,i we have
(37)
Ψα ξ˜α,i (D̃α,i ) = ξ¯α,i (D̄α,i ) and mα,i = m̄α,i .
The proof is straightforward. Let x ∈ X be any element and let ϕx : Vx → Ux ⊂ X
be the analytic open embedding provided by Lemma 2.4. By the construction of the
projection U [[α]] → X , its fiber over x is canonically isomorphic to
U [[α]] ×X n Uxn ×Tx X 0
in which the map Uxn → Tx X is the composite
∼
=
⊂
ave
$x : Uxn −−→ Vxn −−→ Tx X n −−→ Tx X
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Zero dimensional Donaldson–Thomas invariants of threefolds
2167
with the averaging map. Combining the local isomorphism Lemma (Lemma 2.2), the
base change property of the obstruction sheaves (21), the invariance of the obstruction
theory (Lemma 5.3) and the invariance of normal cone (22), we see immediately that to
each stratum S of X [[α]] , the induced map
$x
S ×X n (Ux )n −−→ Tx X
is a submersion. This shows that the standard stratification of X [[α]] induces the
standard stratification of (Ux )[[α]] ×Tx X 0. By using Ux ∼
= Vx and the open inclusion
[[α]]
Vx ⊂ Tx X , the standard stratification of (Tx X)0 also induces the standard stratification
of (Ux )[[α]] ×Tx X 0. Therefore, the restrictions of the standard stratification of (Tx X)[[α]]
0
and of the standard stratification of X [[α]] to V0[[α]] ×X x coincide. Consequently, after
re-indexing C̄α,i , we will have (36).
Then by the base change property of the obstruction sheaves, we automatically have
canonical isomorphism
Ψ∗α W̄α,i ∼
= W̃α,i ;
applying the invariance results of the obstruction sheaf and of obstruction theory, we
get the identity (37).
Once we have these, we immediately see that the section s̃α induced by the isomorphism
[[α]]
∼
Ψ∗α (Ob[[α]]
X ) = ObTX |V [[α]]
0
Ob[[α]]
X
U [[α]] ;
is a smooth section of
over
that it intersects transversally with the normal
[[α]]
cone of ObX and the collection {s̃α } satisfies the conclusions of Lemma 5.6 over
U [[α]] .
To continue, we shall comment on the role of 0 < c 1 and 0 < 1. In choosing
sections tα according to Lemma 5.6, we use the smallness of c to force the resulting
pseudo-cycle δ[n],Q to lie entirely in the 2c–tubular neighborhood of the zero section of
Qn0 /Gr. As to , after picking tα we use to choose pseudo-cycle representatives of
D(tα ) to ensure that the resulting pseudo-cycle δ[n],Q represents a homology class in
H∗ (Gr, Z).
After that, we pick a smooth embedding g : X → Gr to pull the sections tα back to
sections t̃α in Ob[[α]]
TX . By choosing g in general position, we can be sure that the
degree of the discrepancy cycle δ[n],TX constructed using t̃α coincides with the pull
back of δ[n],Q under the induced map (TX)n0 → Qn0 . This time because each D(t̃α ) is a
zero-dimensional pseudo-cycle, they are cycles automatically. Likewise, because δ[n],Q
lies in the 2c–tubular neighborhood of the zero section of Qn0 /Gr, for c sufficiently
small, δ[n],TX lies entirely in V0n .
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Jun Li
The next step is to form the sections s̃α of Ob[[α]]
over U [[α]] ; restrict them to a compact
X
subset K [[α]] ⊂ U [[α]] and then extend the restrictions to all X [[α]] so that the resulting
sections {sα } satisfy the conclusion of Lemma 5.6. Now let ∆n ⊂ X n be the top
diagonal {(x, · · · , x) | x ∈ X} as before and let ∆n,3c be the 3c–tubular neighborhood
of ∆n in X n . Because c is sufficiently small, we can choose a compact K ⊂ X n so that
∆n,3c ⊂ K ⊂ V0n ⊂ X n .
Thus if we choose K [[α]] = X [[α]] ×X n K , then the discrepancy cycle δ[n] constructed
using D(sα ) is entirely contained in ∆n,3c ; thus is contained in U [[α]] ; thus coincide
with Ψ[n] (δ[n],TX ). Here the last statement holds because δ[n],TX ⊂ V0n .
This completes the proof of Lemma 6.1.
7
The case of complex manifolds
To generalize this theorem to cover all compact, smooth three-dimensional complex
manifolds, we first need to define their Hilbert schemes of subschemes and their
Donalson–Thomas invariants. The technique developed in this work readily covers the
case of zero-dimensional invariants.
We now construct the Hilbert scheme of points for such complex manifolds X . We shall
achieve this goal indirectly by quoting the invariance results proved in this paper. We
first cover X by open subsets Uα such that each is realized as an open subset of A3 . By
viewing Uα as an open subset of A3 , we define the Hilbert scheme of n–points
IUα (0, n) ⊂ IA3 (0, n)
be the open (analytic) subscheme of all ξ ∈ IA3 (0, n) whose supports lie in Uα . For any
pair Uα and Uβ , the Lemma 2.2 ensures that the open subscheme
IUαβ (0, n) ⊂ IUα (0, n)
is canonically isomorphic to the open subscheme
IUαβ (0, n) ⊂ IUβ (0, n).
Thus the collection IUα (0, n) glue to form an analytic scheme, the Hilbert scheme of n
points in X :
IX (0, n).
The scheme IX (0, n) comes with the usual obstruction sheaf, obstruction theory and
virtual normal cone. Because of the invariance results stated or proved in this paper, the
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2169
obstruction sheaves Obα of IUα (0, n) glue together to form the obstruction sheaf of
IX (0, n). It can also be defined as the traceless relative extension sheaf of the universal
ideal sheaf of Z ⊂ X × IX (0, n):
Ob = Ext 2π2 (IZ , IZ )0 .
Over each open IUα (0, n), we can find locally free sheaf Eα making the obstruction
sheaf Obα of IUα (0, n) its quotient sheaf; we can also construct its associated normal
cone Cα in the associated vector bundle Eα . Using
Cα ⊂ Eα
and
Eα −→ Obα ,
we can define the notion of smooth sections of Obα and when smooth section sα of
Obα intersects transversally with the normal come. Because such notion is consistent
when restricted to open subsets IUαβ (0, n), we can make sense of smooth sections of
Ob on IX (0, n) and when it intersects transversally with the normal cone of Ob.
We then define the virtual cycle IX (0, n) be the homology class
[D(s)] ∈ H0 (IX (0, n); Z)
represented by the pseudo-cycle D(s) constructed by intersecting the graph of s with
the normal cone in Ob.
To show that the cycle [D(s)] is well-defined, namely it is independent of the choice of
s, we need to show that for different smooth sections s the cycles D(s) are homotopy
equivalent. This is true because we can find a stratification of IX (0, n) and of the cone
so that each stratum is the complement of finitely many closed analytic subvarieties in a
closed analytic variety.
Combined, this proves
Theorem 7.1 Let X be a compact, smooth three dimensional complex manifold. Then
the so constructed Hilbert scheme of points IX (0, n) has a well-defined virtual cycle
[IX (0, n)]vir ∈ H0 (IX (0, n); Z)
that is represented by the cycle D(s) after intersecting a smooth section s transversally
with the normal cone in the obstruction sheaf Ob.
Once the virtual cycle is constructed, then the proof of this paper applies line to line to
IX (0, n) to conclude that
deg[IX (0, n)]vir
is expressible by the same universal expression in its Chern numbers as other projective
threefolds. Thus
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Jun Li
Theorem 7.2 The identity
X
deg[IX (0, n)]vir qn = M(−q)c3 (TX ⊗KX )
n
holds for all compact, smooth three dimensional complex manifolds.
The Hilbert scheme of ideal sheaves of curves for any complex manifold X can also
be defined. In case X has dimension three, one can also define its virtual cycle and its
Donaldson–Thomas series, along the lines of the work [8].
Acknowledgement The authors is partially supported by NSF grants DMS-0200477
and DMS-0244550.
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Department of Mathematics, Stanford University
Stanford, CA 94305, USA
[email protected]
Proposed: Jim Bryan
Seconded: Lothar Goettsche, Eleny Ionel
Geometry & Topology 10 (2006)
Received: 27 April 2006
Accepted: 10 October 2006